college of engineering, mathematics and physical sciences
TRANSCRIPT
College of Engineering, Mathematics and Physical Sciences
A Simulation-Optimization Model to Study the Control of
Seawater Intrusion in Coastal Aquifers
Submitted by
Hany Farhat Abd-Elhamid
To the University of Exeter as a thesis for the degree of
Doctor of Philosophy in Water Engineering
December 2010
This thesis is available for library use on the understanding that it is copyright material
and that no quotation from the thesis may be published without proper
acknowledgement.
I certify that all material in this thesis which is not my own work has been identified and
that no material has previously been submitted and approved for the award of a degree
by this or any other university.
Hany F. Abd-Elhamid
A Simulation-Optimization Model to Study the Control of
Seawater Intrusion in Coastal Aquifers
Ph.D. in Water Engineering
University of Exeter
December 2010
Acknowledgment
i
ACKNOWLEDGEMENTS
Firstly, I thank my God for giving me the strength and endurance to complete this
research.
I would like to express my profound gratitude to my supervisor Dr. Akbar Javadi for his
great support, invaluable guidance, continuous encouragement, unlimited help and great
efforts he presented to achieve this research project.
Many thanks to all the members of Computational Geomechanics Group for their help
and support during my study. Also, I would like to thank all the staff at the College of
Engineering, Mathematics and Physical Sciences in Exeter University to all their help
during my time in Exeter University.
I also want to thank all the staff of Water and Water Structure Engineering Department,
Faculty of Engineering, Zagazig University, Zagazig, Egypt to all their help and support
before and during my scholarship and for funding this research.
I wish to express my gratitude to all members of my family (my Mum, brothers and
sister) who supported me and encouraged me to continue my work. I am also deeply
grateful to my wife for her love, support, prayers, patience, and encouragement during
my study. I also want to thank my friends in Egypt for their prayers and their continuous
encourage pursuing my goals.
Abstract
ii
ABSTRACT
Groundwater contamination is a very serious problem as it leads to the depletion of water
resources. Seawater intrusion is a special category of groundwater contamination that
threatens the health and possibly lives of many people living in coastal areas. The focus of
this work is to develop a numerical model to study seawater intrusion and its effects on
groundwater quality and develop a control method to effectively control seawater intrusion.
Two major approaches are used in this study: the first approach is the development of a
finite element model to simulate seawater intrusion; the second is the development of a
simulation-optimization model to study the control of seawater intrusion in coastal aquifers
using different management scenarios. The simulation-optimization model is based on the
integration of a genetic algorithm optimization technique with the transient density-
dependent finite element model developed in this research.
The finite element model considers the coupled flow of air and water and solute transport in
saturated and unsaturated soils. The governing differential equations include two mass
balance equations of water and air phases and the energy balance equation for heat transfer,
together with a balance equation for miscible solute transport. The nonlinear governing
differential equations are solved using the finite element method in the space domain and a
finite difference scheme in the time domain. A two dimensional finite element model is
developed to solve the governing equations and provide values of solute concentration,
pore water pressure, pore air pressure and temperature at different points within the region
at different times. The mathematical formulation and numerical implementation of the
model are presented. The numerical model is validated by application to standard examples
from literature followed by application to a number of case studies involving seawater
intrusion problems. The results show good agreement with previous results reported in the
literature. The model is then used to predict seawater intrusion for a number of real world
case studies. The developed model is capable of predicting, with a good accuracy, the
intrusion of seawater in coastal aquifers.
Abstract
iii
In the second approach, a simulation-optimization model is developed to study the control
of seawater intrusion using three management scenarios: abstraction of brackish water,
recharge of fresh water and combination of abstraction and recharge. The objectives of
these management scenarios include minimizing the total costs for construction and
operation, minimizing salt concentrations in the aquifer and determining the optimal
depths, locations and abstraction/recharge rates for the wells. Also, a new methodology is
presented to control seawater intrusion in coastal aquifers. In the proposed methodology
ADR (abstraction, desalination and recharge), seawater intrusion is controlled by
abstracting brackish water, desalinating it using a small scale reverse osmosis plant and
recharging to the aquifer. The simulation-optimization model is applied to a number of case
studies. The efficiencies of three different scenarios are examined and compared. Results
show that all the three scenarios could be effective in controlling seawater intrusion.
However, ADR methodology can result in the lowest cost and salt concentration in aquifers
and maximum movement of the transition zone towards the sea. The results also show that
for the case studies considered in this work, the amount of abstracted and treated water is
about three times the amount required for recharge; therefore the remaining treated water
can be used directly for different proposes. The application of ADR methodology is shown
to be more efficient and more practical, since it is a cost-effective method to control
seawater intrusion in coastal aquifers. This technology can be used for sustainable
development of water resources in coastal areas where it provides a new source of treated
water. The developed method is regard as an effective tool to control seawater intrusion in
coastal aquifers and can be applied in areas where there is a risk of seawater intrusion.
Finally, the developed FE model is applied to study the effects of likely climate change and
sea level rise on seawater intrusion in coastal aquifers. The results show that the developed
model is capable of predicting the movement of the transition zone considering the effects
of sea level rise and over-abstraction. The results also indicate that the change of water
level in the sea side has a significant effect on the position of the transition zone especially
if the effect of sea level rise is combined with the effect of increasing abstraction from the
aquifer.
Table of contents
iv
TABLE OF CONTENTS
ACKNOWLEDGEMENTS…………………………………...……………………..….….. i
ABSTRACT………………………………………………………………..………..….….. ii
TABLE OF CONTENTS……………………………………………….…………..….….. iv
LIST OF FIGURES……………………………………………………...…………..….….. x
LIST OF TABLES……………………………………………………..……...…..….….. xiii
LIST OF PUPLICATIONS…………………………………...……….…………..….….. xiv
LIST OF ABBREVIATIONS……………………………………………………..….….. xvi
LIST OF SYMBOLS ………….……………………………………..………..….…….. xvii
Chapter 1: INTRODUCTION………………………………………………...……...….. 1
1.1 General……………………………………………………………………………...….. 1
1.2 Groundwater contamination …………………………………………….…………….. 2
1.3 Saltwater intrusion problem………………………………………………………...….. 4
1.4 Saltwater intrusion control methods………………………………………………...….. 5
1.5 Proposed methodology to control saltwater intrusion …..……...……….…………….. 6
1.6 Effects of climate change and sea level rise on saltwater intrusion …….…………….. 7
1.7 Objectives of the study ………………………………………………….…………….. 8
1.8 Outline of the thesis …………………………………………………….……..…..….. 9
Chapter 2: SALTWATER INTRUSION INVESTIGATION AND CONTROL
METHODS…………………...…………..…………………………...... 11
2.1 Introduction………………………………………………………………………..….. 11
2.2 Main causes of saltwater intrusion in coastal aquifers ……………………...……….. 12
2.3 Saltwater intrusion investigation methods.…......................................................…….. 13
2.3.1 Geophysical investigations….………………………………...…………….. 13
2.3.2 Geochemical investigations …..…………..……….……………...……….. 16
2.3.3 Experimental studies …..…………..……….………………………..…….. 17
2.3.4 Mathematical methods …..…………..……….……………………...…….. 18
Table of contents
v
2.4 History of saltwater intrusion modelling …………………………………………….. 19
2.4.1 Sharp-interface or diffusive interface models ….…………...….…….…….. 20
2.4.1.1 Sharp-interface models ….…………….…….………………...….. 20
2.4.1.2 Diffusive interface models ….…………….…….…………….….. 22
2.4.2 Steady-state and transient flow models ….…………………...…....……….. 26
2.4.3 Saturated and unsaturated flow models …..……………………...……..….. 27
2.4.4 Recent computer codes to simulate saltwater intrusion ………………..….. 28
2.5 Saltwater intrusion control methods …………………………………………...…..... 31
2.5.1 Reduction of pumping rates ……………………………………....….…….. 32
2.5.2 Relocation of pumping wells ………………………………..……………... 33
2.5.3 Subsurface barriers ……………………………………………..…………... 35
2.5.4 Natural recharge ……………………………………………...…………..… 36
2.5.5 Artificial recharge …………………………………….……..……….…….. 37
2.5.6 Abstraction of saline water …………………………………..…………….. 41
2.5.7 Combination techniques ……………………………………..……………... 42
2.6 A new methodology to control saltwater intrusion (ADR) ......………………..…...... 44
2.6.1 The benefits of ADR methodology ………………………….…..…………. 45
2.6.2 Desalination of brackish groundwater using RO……..…………...……..…. 46
Chapter 3: GOVERNING EQUATIONS OF COUPLED FLUID FLOW AND
SOLUTE TRANSPORT …………………....………..…………………….50
3.1 Introduction………………………………………………………………………..….. 50
3.2 History of coupled fluid flow and solute transport ………………….……...……….. 51
3.2.1 Fluid flow in saturated/unsaturated soils ………………………………….. 51
3.2.2 Energy transport ………………………………………………………..….. 54
3.2.3 Solute transport in saturated/unsaturated soils ……………..………….….. 55
3.2.4 Coupling mechanisms ………………………………..………….……..….. 59
3.3 The governing equation for the moisture transfer ………………………...………..... 61
3.3.1 Water flow ……………………………………………………….…...…….. 63
3.3.2 Water vapour flow….……………………………………………………….. 66
3.3.3 Mass balance equation for the moisture transfer…………………………..... 72
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vi
3.4 The governing equation for air transfer ……………………...…………...………..... 75
3.5 The governing equation for heat transfer …………………...………….....………..... 81
3.6 The governing equation for solute transport ……….………...…………...………..... 87
3.6.1 Net rate of solute inflow the control volume ….…………...…...………..... 88
3.6.1.1 Advection ….………………………………………….…...…….. 89
3.6.1.2 Diffusion ….…….……………………………………………….. 89
3.6.1.3 Mechanical Dispersion …………………………………..……..... 90
3.6.2 Net rate of solute production within the control volume ….……………..... 92
3.6.3 Governing equation of solute transport in saturated soil …………..……..... 94
3.6.4 Governing equation of solute transport in unsaturated soil ………..……..... 95
3.7 The governing differential equations ……….………...…………...………...……..... 98
Chapter 4: NUMERICAL SOLUTION OF GOVERNING EQUATIONS OF
COUPLED FLUID FLOW AND SOLUTE TRANSPORT ….....………...99
4.1 Introduction………………………………………………………………..…..…..….. 99
4.2 Solution methods for differential equations ……………………………...….....…..... 99
4.2.1 Analytical methods ……………………………………………..…...…….. 100
4.2.2 Numerical methods ….……………………………………………...…….. 101
4.2.2.1 Finite difference method ………………………………..……..... 101
4.2.2.2 Finite element method ……………………………………..…..... 101
4.3 Numerical solution of the governing equations ……….…...……………....……..... 103
4.3.1 Spatial discretization of the moisture transfer governing equation……....... 104
4.3.2 Spatial discretization of the air transfer governing equation……….……... 110
4.3.3 Spatial discretization of the heat transfer governing equation………….…. 114
4.3.4 Spatial discretization of the solute transport governing equation…….…… 119
4.3.4.1 Solute transport in saturated soils ……………………………..... 119
4.3.4.2 Solute transport in unsaturated soils ……………….…...……..... 123
4.3.5 Temporal discretization of the coupled fluid flow and heat transfer ……... 127
4.3.6 Temporal discretisation of solute transport …………………...................... 129
4.4 Coupling of fluid flow and solute transport equations …............................................ 131
Table of contents
vii
Chapter 5: VALIDATION AND APPLICATION OF THE SIMULATION
MODEL……………………………………………………………………...133
5.1 Introduction…………………………………………………………………..…..….. 133
5.2 Calibration, validation, and verification of the model. ……………….…………..... 133
5.3 Numerical results. …………………………………………………………….…..... 136
5.3.1 Example (1): Transient groundwater flow ………………………….…….. 136
5.3.2 Example (2): Transient solute transport …………………………………... 138
5.3.3 Example (3): Saltwater intrusion in confined aquifer (Henry’s problem)… 141
5.3.3.1 Steady-state constant dispersion …………………………..…..... 143
5.3.3.2 Steady-state variable dispersion …………………………...…..... 145
5.3.3.3 Transient constant dispersion …….………………………..…..... 146
5.3.3.4 Transient variable dispersion ………………………….…..…..... 147
5.3.3.5 Effect of variability in dispersion and density of SWI……..…..... 147
5.4 Case studies. …………………………………………………………………..…..... 149
5.4.1 Case (1): A hypothetical case study …………………………..…….…….. 149
5.4.2 Case (2): Madras aquifer, India ……………………………….…………... 152
5.4.3 Case (3): Biscayne aquifer, Florida, USA …...…………………………..... 154
5.4.4 Case (4): Gaza Aquifer, Palestine ……...………………………………..... 157
5.5 Summary. ……………………………………………………………………..…..... 164
Chapter 6: DEVELOPMENT AND APPLICATION OF SIMULATION-
OPTIMIZATION MODEL……………………………………………..165
6.1 Introduction…………………………………………………………………..…..….. 165
6.2 Optimization techniques ……..……………………………………….…………..... 166
6.3 Genetic algorithm (GA). ……………………………………………………...…..... 168
6.3.1 GA in groundwater management ………………………………..….…….. 168
6.3.2 GA in water resources management …………………………….………... 169
6.3.3 GA in seawater intrusion management …………………………….……... 169
6.4 Simulation-Optimization model …………………………………………………….. 173
6.4.1 Simulation model …………………………………….……………..…….. 175
6.4.2 Genetic Algorithm Optimization model …………………………….…….. 177
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viii
6.4.3 Integration of the simulation and optimization models ……..…..…..…….. 178
6.5 Formulation of the management models ……………………………………...…….. 181
6.5.1 Management model 1: (Abstraction wells) ……….….……………..…….. 182
6.5.2 Management model 2: (Recharge wells) ……...…………………….…….. 184
6.5.3 Management model 3: (Abstraction and Recharge wells)……………..….. 185
6.6 Application of the simulation-optimization model …………….…………......…….. 186
6.6.1 A hypothetical case study (Henry’s problem) ……….……………..…….. 186
6.6.1.1 Application of management model 1 ………………………….... 189
6.6.1.2 Application of management model 2 ………………………….... 190
6.6.1.3 Application of management model 3 ………………………….... 190
6.6.1.4 Discussion ………………………………….………………….... 192
6.6.2 Real world case study (Biscayne aquifer, Florida, USA) ……..………….. 194
6.6.2.1 Application of management model 1 ………………………….... 197
6.6.2.2 Application of management model 2 ………………………….... 197
6.6.2.3 Application of management model 3 ………………………….... 197
6.6.2.4 Discussion ………………………………….………………….... 199
6.7 Summary ……………………….......................................................................…….. 200
Chapter 7: EFFECT OF CLIMATE CHANGE AND SEA LEVEL RISE ON
SALTWATER INTRUSION …..………………………………………...202
7.1 Introduction…………………………………………………………………..…..….. 202
7.2 Climate change. ……………………………………………………….…………..... 203
7.3 Sea level rise. …………………………………………………………….…...…..... 205
7.3.1 Effects of sea level rise on coastal areas…………………….…...….…….. 206
7.3.1.1 Erosion and inundation ………………………………………..... 208
7.3.1.2 Saltwater intrusion …………………………………………….... 209
7.3.2 Methods of controlling the effects of sea level rise on coastal areas …...… 210
7.3.2.1 Control of erosion and inundation …………………………...….. 210
7.3.2.2 Control of saltwater intrusion ………………...……………...….. 211
7.4 Simulation of saltwater intrusion considering sea level rise ……..…………...…..... 212
7.5 Application of SUFT to investigate the effect of sea level rise ……...…………….. 213
7.5.1 Case (1): A hypothetical case study ……….…..………….………...…..... 213
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ix
7.5.2 Case (2): Biscayne aquifer, Florida, USA …………………………........... 216
7.5.3 Case (3): Gaza Aquifer, Palestine …………………………………........... 219
7.6 Summary ……………………….......................................................................…….. 222
Chapter 8: CONCLUSIONS AND RECOMMENDATIONS …………………….....223
8.1 Conclusions…………………………………………………………………..…..….. 223
8.2 Recommendations for future work ……………..…………………….…………..... 225
REFERENCES ………………………………………………...………………………. 226
List of figures
x
LIST OF FIGURES
Figure 1.1 Schematic diagram of hydrologic cycle…………...………..………………….. 1
Figure 1.2 Main groundwater contaminant pathways and sources…………...……..…….. 3
Figure 1.3 A cross section with interface under natural conditions…………...….……….. 4
Figure 1.4 The proposed methodology (ADR) to control saltwater intrusion…………….. 6
Figure 2.1 Hydrostatic equilibrium between fresh and saline water……………….…….. 21
Figure 2.2 Location of transition zone………………………………………………….... 23
Figure 2.3 Sketch for subsurface barrier….……………………………..……………….. 35
Figure 2.4 Sketch for natural recharge process………………………………..…………. 36
Figure 2.5 Sketch of recharge and abstraction wells……………………..………………. 38
Figure 2.6 Sketch of the new methodology ADR …………………………….…………. 44
Figure 2.7 Schematic sketch layout of RO desalination plant………………..……….…. 48
Figure 3.1 Control volume of solute transport through pours media…………………….. 88
Figure 4.1 Summary of the solution methods for differential equations……………….. 100
Figure 5.1 Boundary and initial conditions of example (1) ………………………...….. 137
Figure 5.2 Head distribution at different times in the reservoir……………………….... 137
Figure 5.3 Comparison between the current model and Wang and Anderson
results for head distribution in the reservoir…………………………………………….. 138
Figure 5.4 Boundary and initial conditions of example (2) ………………………...….. 139
Figure 5.5 Solute concentrations distribution at different times in the reservoir……….. 139
Figure 5.6 Comparison between the current model and Wang and Anderson
results for solute concentrations distribution in the reservoir………………………….... 140
Figure 5.7 Boundary conditions of Henry’s problem……………………………..……. 141
Figure 5.8 0.5 Isochlor for steady-state constant dispersion (D = 6.6x10-6) ………..….. 144
Figure 5.9 0.5 Isochlor for steady-state constant dispersion (D = 2.31x10-6) ….…....…. 144
Figure 5.10 0.5 Isochlor for steady-state variable dispersion…………………….…….. 145
Figure 5.11 0.5 Isochlor for transient constant dispersion……………………..……….. 146
Figure 5.12 0.5 Isochlor for transient variable dispersion………………….….….…….. 147
Figure 5.13 0.5 Isochlor for constant and variable dispersion coefficient……….….…. 148
List of figures
xi
Figure 5.14 0.5 Isochlor for constant and variable density……………….……………. 148
Figure 5.15 Problem description of unconfined coastal aquifer
with fresh water channel at land side……………….………………………………….... 150
Figure 5.16 Boundary conditions of the hypothetical case……………….…………….. 151
Figure 5.17 0.5, 0.25, 0.1, 0.05 and 0.005 Isochlor contours of the hypothetical case…. 151
Figure 5.18 Boundary conditions of Madras aquifer……………….……….………….. 153
Figure 5.19 35, 17.5, 10, 5 and 2.0 equi-concentration lines in Madras aquifer….…….. 154
Figure 5.20 Schematic sketch and boundary conditions of Biscayne aquifer, Florida…. 155
Figure 5.21 1, 0.75, 0.5 and 0.25 isochlor counters in Biscayne aquifer, Florida………. 156
Figure 5.22 Location map of Gaza strip……………………………………………….... 157
Figure 5.23 Isolines of TDS concentration (kg/m3) calculated by (SUFT) along
Jabalya …………………………………………………………………..………………. 160
Figure 5.24 Isolines of TDS concentration (kg/m3) calculated by (SUFT) along
Wadi Gaza.…………………..…………………………………………..………………. 160
Figure 5.25 Isolines of TDS concentration (kg/m3) calculated by (SUFT) along
Khan Yunis ……………………………….……………………………..………………. 161
Figure 5.26 Isolines of TDS concentration (kg/m3) calculated by SEAWAT along
Khan Yunis (Qahman and Larabi, 2006)……..…………………………………………. 162
Figure 5.27 Isolines of TDS concentration (kg/m3) calculated by SEAWAT along
Jabalya (Qahman and Larabi, 2006) ……..…...……...…………………………………. 162
Figure 5.28 Extant of TDS concentration calculated by SEAWAT
(Qahman and Larabi, 2006)……..…………………...…………………………………. 163
Figure 6.1 Flow chart of the solution algorithm…………………………………...……. 176
Figure 6.2 Flowchart for the genetic algorithm…………………………………………. 179
Figure 6.3 Flow chart of the schematic steps of the simulation-optimization model…... 180
Figure 6.4 Schematic sketch for potential locations and depths
for the abstraction and recharge wells………………………………...…………………. 181
Figure 6.5 0.5 isochlor for the hypothetical case study……………………………….... 187
Figure 6.6 Schematic sketch for potential locations and depths
for the abstraction and recharge wells of the hypothetical case study………………..…. 187
Figure 6.7 0.5 isochlor distribution from simulation-optimization
models for the hypothetical case……………………………………………………….... 191
List of figures
xii
Figure 6.8 Comparison between total costs of model 1, 2 and 3
for the hypothetical case study………………………………………………………..…. 193
Figure 6.9 Comparison between total salt concentrations remaining in the aquifer
of model 1, 2 and 3 for the hypothetical case study…………………………………..…. 193
Figure 6.10 Comparison between total abstraction/recharge rates
of model 1, 2 and 3 for the hypothetical case study…………………………..…………. 194
Figure 6.11 Isochlor counters in Biscayne aquifer, Florida………………………….…. 195
Figure 6.12 Schematic sketch for potential locations and depths
for the abstraction and recharge wells of Biscayne aquifer, Florida………………...…... 195
Figure 6.13 0.5 isochlor distribution from simulation-optimization models for
Biscayne aquifer, Florida…………………………………………………..……………. 198
Figure 6.14 Comparison between total costs of model 1, 2 and 3
for Biscayne aquifer, Florida………………………………………………………….…. 199
Figure 6.15 Comparison between total salt concentrations remaining in the aquifer
of model 1, 2 and 3 for Biscayne aquifer, Florida………………………………………. 199
Figure 6.16 Comparison between total abstraction/recharge rates
of model 1, 2 and 3 for Biscayne aquifer, Florida………………………………………. 200
Figure 7.1 Scenarios for estimating average global temperature of earth (IPCC, 2001)...204
Figure 7.2 Scenarios for estimating global average sea level rise (IPCC, 2001)….….... 206
Figure 7.3 Effect of sea level rise on abstraction wells…………………………………. 207
Figure 7.4 The relationship between sea level rise and saltwater intrusion…………….. 210
Figure 7.5 0.5 isochlor for the hypothetical case study (no SLR) ………………………. 214
Figure 7.6 0.5 isochlor for the hypothetical case study (with SLR) ……………………. 215
Figure 7.7 Isochlor contours in Biscayne aquifer, Florida (no SLR) ………………..…. 216
Figure 7.8 Isochlor contours in Biscayne aquifer, Florida (with SLR, 50, 100 year)..…. 217
Figure 7.9 0.5 Isochlor contours in Biscayne aquifer, Florida using three scenarios..….. 218
Figure 7.10 Isolines of TDS concentration (kg/m3) along Jabalya (no SLR)...………..... 219
Figure 7.11 Isolines of TDS concentration (kg/m3) along Jabalya (Scenario 1)...…….... 220
Figure 7.12 Isolines of TDS concentration (kg/m3) along Jabalya (Scenario 2)...…........ 221
Figure 7.13 Isolines of TDS concentration (kg/m3) along Jabalya (Scenario 3)...…........ 221
List of tables
xiii
LIST OF TABLES
Table 2.1 The water type based on TDS content ................................................................ 13
Table 5.1 The parameters used in Henry’s problem.......................................................... 142
Table 6.1 Summary of the upper and lower constraints for the hypothetical case study.. 188
Table 6.2 Summary of the values of cost.......................................................................... 189
Table 6.3 Summary of the results obtained from the simulation-optimization
models for the hypothetical case study.............................................................................. 191
Table 6.4 Summary of the upper and lower of constraints for Biscayne aquifer,
Florida ............................................................................................................................... 196
Table 6.5 Summary of the results obtained from the simulation-optimization models
for Biscayne aquifer, Florida.............................................................................................. 198
List of publications
xiv
LIST OF PUBLICATIONS
These publications were completed during the author’s period of registration to the higher
degree program.
Refereed Journals
H. F. Abd-Elhamid and A. A. Javadi (2010). “A Cost-effective Method to Control
Seawater Intrusion in Coastal Aquifers”, Journal of Water Resources Management
(in press).
A.A. Javadi, H. F. Abd-Elhamid and R. Farmani (2010). “A Simulation-
Optimization Model to Control Seawater Intrusion in Coastal Aquifers Using
Abstraction/Recharge Wells”, International Journal for Numerical and Analytical
Methods in Geomechanics (in press).
H. F. Abd-Elhamid and A. A. Javadi (2010). “Impact of Sea Level Rise and Over-
pumping on Seawater Intrusion in Coastal Aquifers”, Journal of Water and Climate
Change (in press).
H. F. Abd-Elhamid and A. A. Javadi (2010). “A Density-Dependant Finite Element
Model for Analysis of Saltwater Intrusion in Coastal Aquifers”, Journal of
Hydrology (in press).
H. F. Abd-Elhamid and A. A. Javadi (2009). “Effects of climate change and
saltwater on coastal cities”, Climate change, Barcelona Metropolis, Spain, No.75,
pp: 78-80.
List of publications
xv
Refereed Conferences
A.A. Javadi and H.F. Abd-Elhamid (2010). “Optimal Management of Groundwater
Resources in Coastal Aquifers Subjected to Seawater Intrusion”, Proceeding of
(ISMAR7), Oct 9-13 Abu Dubai, UAE.
H.F. Abd-Elhamid and A.A. Javadi (2010). “A Simulation-Optimization Model to
Study the Control of Seawater Intrusion in Coastal Aquifers Using ADR
Methodology”, Proceeding of 21st SWIM, June 21-26, 2010, Azores, Prtogal.
H.F. Abd-Elhamid and A.A. Javadi (2008b). “An investigation into control of
saltwater intrusion considering the effects of climate change and sea level rise”,
Proceeding of 20th SWIM, June 23-27, 2008, Naples, Florida, USA.
A.A. Javadi and H.F. Abd-Elhamid (2008). “A New Methodology to Control
Saltwater Intrusion into Coastal Aquifers” Proceeding of first national seminar of
geotechnical problems of irrigation and drainage networks, May 2008, Tehran, Iran.
H.F. Abd-Elhamid and A.A. Javadi (2008a). “Mathematical models to control
saltwater intrusion in coastal aquifer”, Proceeding of GeoCongress 2008, March 9-
12, 2008, New Orleans, Louisiana, USA.
List of abbreviations
xvi
LIST OF ABBREVIATIONS
ADR Abstraction, Desalination and Recharge
ANN Artificial Neural Network
FE Finite Element
FED Finite Difference Method
FEM Finite Element Method
FVUM Finite Volume Unstructured Mesh
GA Genetic Algorithm
GIS Geographic Information System
IPCC Intergovernmental Panel on Climate Change
NE Number of Elements
NN Number of Nodes
RO Reverse Osmosis
SEWAT Simulation of Three-Dimensional Variable-Density Ground-Water Flow and
Transport
S/O Simulation/Optimization
SLR Sea Level Rise
SUFT Saturated/Unsaturated Fluid flow and solute Transport
SUTRA Saturated-Unsaturated Variable-Density Ground-Water Flow with Solute or
Energy Transport
SWI Saltwater Intrusion
TDS Total Dissolved Solids
List of symbols
xvii
LIST OF SYMBOLS
c Solute Concentration
rC Courant number
psC Specific heat capacities of solid particles
pwC Specific heat capacities of water
pvC Specific heat capacities of vapour
pdaC Specific heat capacities of dry air
CA Cost of abstraction
CT Cost of treatment
CDW Cost of installation/drilling of well
CR Cost of recharge
CPW Price of water
atmsD Molecular diffusivity of vapour through air
mD Molecular diffusion coefficient in the porous medium
amD Coefficient of air molecular diffusion
moD Molecular diffusion coefficient in aqueous solution
DA Depth of abstraction well
DR Depth of recharge well
e Void ratio
F Source or sink term
f Objective function
g Gravitational acceleration
H Henry’s constant
aH Henry’s volumetric coefficient of solubility
cH Heat capacity of the soil at the reference temperature
h Relative humidity
wh Hydraulic head
List of symbols
xviii
sh Depth of interface below mean sea level
fh Height of the potentiometric surface above the mean sea level
i Iteration number
wK Hydraulic conductivity of water
aK Unsaturated conductivity of air
ak Effective permeability of air
wk Effective permeability of water
fk Constant for the forward reaction
rk Constant for reverse reaction
dk Equilibrium distribution coefficient
L Latent heat of vaporisation
N Shape function
n Porosity
eP Peclet Number
Q Heat flux per unit area
QA Abstraction rate
QR Recharge rate
R Retardation factor
vR Specific gas constant for water vapour
wR Residual error introduced by the approximation functions
aR Residual errors introduced by the approximation functions
TR Residual error introduced by the approximation functions
cR Residual errors introduced by the approximation function
daR Specific gas constant of dry air
S Salinity
s Suction
aS Degree of air saturation in unsaturated soil
wS Degree of water saturation in unsaturated soil
List of symbols
xix
rs Suction at the reference temperature
T Absolute temperature
rT Reference temperature
t Time step
wu Pore water pressure
au Pore air pressure
vu Pore water vapour pressure
dau Pore dry air pressure
wv Velocity of water
av Velocity of the air
vv Velocity of the water vapour
xwv Component of the pore water velocity in x direction
ywv Component of the pore water velocity in y direction
v Magnitude of the water velocity
xav Component of the pore air velocity in x direction
yav Component of the pore air velocity in y direction
v Magnitude of the air velocity
wnv̂ Approximation of water velocity normal to the boundary surface
vdv̂ Approximation of diffusive vapour velocity normal to the boundary surface
vav̂ Approximation of pressure vapour velocity normal to the boundary surface
fnv̂ Approximation of velocity of free dry air
anv̂ Approximation of velocity of dissolved dry air
vV Mass flow factor
z Elevation above the datum (positive upwards)
z The unit normal oriented downwards in the direction of the force of gravity
s Saltwater density
f Freshwater density
List of symbols
xx
w Density of water
a Density of air
v Density of water vapour
v Vapour density gradient
b Bulk density of the porous media
0 Saturated soil vapour density
da Density of dry air
s Density of solid particles
w Unit weight of water
w Dynamic viscosity of water
a Absolute viscosity of air
Volumetric moisture content in unsaturated soil
w Volumetric water content
a Volumetric air content
v Volumetric vapour content
s Saturated volumetric water content
r Surface energy at the reference temperature
Surface energy at the given temperature T
wT Transverse dispersivity for water phase
Lw Longitudinal dispersivity for water phase
aT Transverse dispersivity for air phase
aL Longitudinal dispersivity for air phase
w The thermal conductivity of water
a Thermal conductivity of air
v Thermal conductivity of vapour
T Intrinsic thermal conductivity of soil
v Vapour torousity factor
List of symbols
xxi
Capillary potential
Domain region e Element domain
An empirical correction factor
Decay constant for the solute e Element boundary surface
Chapter (1) Introduction
1
Introduction
1.1 General
Groundwater is a far more important resource than is often realized. It represents about 22
percent of all fresh water on the earth; polar ice represents 77 percent while other
freshwater in rivers and lakes represent about 0.3 percent (Bear et al., 1999). Groundwater
constitutes an important water resource, supplying water for domestic, industry and
agriculture. One-third of the world’s drinking water is provided by groundwater. But as it is
often the case with critical resources, groundwater is not always available when and where
needed, especially in water-short areas where heavy use has depleted underground reserves.
Groundwater or subsurface water is a term used to denote all the waters found beneath the
surface of the ground and is considered as part of the hydrologic cycle (Figure 1.1).
http://dardel.info/EauAnglais.html
Figure 1.1 Schematic diagram of hydrologic cycle
Chapter (1) Introduction
2
Population growth and continuous development require large quantities of water. It is a
great challenge to secure water demand, while the available water resources are nearly
constant. This requires protecting available resources from pollution including saltwater
intrusion and other contaminants, which deplete the current resources. The concentrations
of people in the coastal regions and the increased related activities have led to an increase
in the abstraction of groundwater reserves. Over-abstraction from coastal aquifers has led to
the movement of saltwater toward aquifers and increased the salinity of groundwater. High
salinity of groundwater limits its usage for irrigation and drinking purposes unless
desalinated or mixed with lower salinity water. The protection of groundwater resources
becomes an essential matter under the conditions of increasing demands and decreasing the
available resources.
1.2 Groundwater contamination
Groundwater quality is an important issue in the development and management of water
resources. In fact, with the increasing demand for water in most parts of the world and with
the intensification of water utilization, the quality problem becomes the limiting factor in
the development and management of water resources in many parts of the world. The
quality of both surface and groundwater resources deteriorates as a result of pollution.
Special attention should be devoted to the pollution of aquifers. Although it seems that
groundwater is more protected than surface water against pollution, it is still subject to
pollution; and when this occurs restoration to the original (non-polluted state) is very
difficult and costly. Groundwater pollution is usually traced back to four sources (Bear,
1979):
1- Environmental: this type of pollution is due to the characteristics of the environment
through which the flow of groundwater takes place. For example, flow through
carbonate rock, seawater intrusion and invasion by brackish water from adjacent aquifer.
2- Domestic: domestic pollution may be caused by accidental breaking of sewers,
percolation from septic tanks, rain infiltrating through sanitary landfills, acid rains,
artificial recharge using sewage water after being treated to different levels and
biological contaminants (e.g., bacteria and viruses).
Chapter (1) Introduction
3
3- Industrial: industrial pollution may come from sewage disposal, which contains heavy
metals, non-deteriorating compounds and radioactive materials.
4- Agriculture: this is due to irrigation water and rain water dissolving and carrying
fertilizers, salts, herbicides, pesticides, etc., as they infiltrate through the ground surface
and replenish the aquifer.
The main pathways and sources of groundwater pollutants are shown in Figure 1.2.
Figure 1.2 Main groundwater contaminant pathways and sources
Saltwater intrusion is one of the most wide-spread and important processes that degrades
groundwater quality by raising salinity to levels exceeding acceptable drinking water and
irrigation standards, and endangers future exploitation of coastal aquifers. Salinization of
groundwater is considered one of the main sources of pollution. The sources of
groundwater salinization include: infiltration of saline wastes, return flow from agriculture,
evapotranspiration and seawater intrusion.
http://www.euwfd.com/html/groundwater.html
Chapter (1) Introduction
4
1.3 Saltwater intrusion problem
Coastal zones are among the most densely populated areas in the world with over 70% of
the world’s population. These regions face serious hydrological problems such as: scarcity
of fresh water, contamination of groundwater and seawater intrusion. The growing in global
population and raising standards of living have increased water demands and pumping from
aquifers. Excessive pumping has led to a dramatic increase in saltwater intrusion problems.
In coastal aquifers a hydraulic gradient exists toward the sea which leads to flow of the
excess fresh water to the sea. Owing to the presence of seawater in the aquifer formation
under the sea bottom, a zone of contact is formed between the lighter freshwater flowing to
the sea and the heavier, underlying, seawater (Figure 1.3). Fresh water and seawater are
actually miscible fluids and therefore the zone of contact between them takes the form of a
tranzition zone caused by hydrodynamic dispersion. Across this zone, the density of the
mixed water varies between that of fresh water to that of seawater. Under certain conditions
the width of this zone is relatively small, when compared with the thickness of the aquifer,
so that the boundary can be considered as a sharp interface that separates the two regions
occupied by the two fluids, in this case the two fluids are assumed to be immiscible. But, if
the transition zone is wide this assumption becomes invalid (Bear, 1979).
Figure 1.3 A cross section with interface under natural conditions
Water Table
Saline groundwater
Fresh groundwater Sea
Ground surface
Sea level
Chapter (1) Introduction
5
Under natural undisturbed conditions in a coastal aquifer, fresh water flows to the sea. By
excessive pumping from a coastal aquifer, the water table or piezometric surface could be
lowered to the extent that the piezometric head in the fresh water body becomes less than in
the adjacent seawater wedge, and the interface could start to advance inland until a new
equilibrium is reached. This phenomenon is called seawater intrusion or encroachment. As
the interface advances, the transition zone widens. When the interface reaches the inland
pumping well, the well becomes contaminated. When the pumping takes place in a well
located above the interface, upconing towards the pumping well could occur. Unless the
rate of pumping is carefully controlled, seawater will eventually enter the pumped well and
the wells may be abandoned. Seawater intrusion leads to the depletion of groundwater
resources. Mathematical models can help in understanding the mechanism of saltwater
intrusion in coastal aquifers. Coupled fluid flow and solute transport models have been
used to simulate saltwater intrusion and to predict the solute concentrations in coastal
aquifers. These models can be used to investigate movement and extension of saltwater into
coastal aquifers.
1.4 Saltwater intrusion control methods
A number of methods have been used to control seawater intrusion to protect groundwater
reserves in coastal aquifers. Todd (1974) presented various methods of preventing seawater
from contaminating groundwater sources including: reduction of abstraction rates,
relocation of abstraction wells, subsurface barriers, natural recharge, artificial recharge, and
abstraction of saline water. Extensive research has been carried out to investigate saltwater
intrusion in coastal aquifers. However, only few models have been developed to study the
control of saltwater intrusion. These models use one or more of the previous measures to
study the control of saltwater intrusion. Many limitations of the previous methods have
been reported in the literature such as: the source and the cost of fresh water and
applicability of such methods (Abd-Elhamid and Javadi, 2008a). Until now there is no
complete solution for this problem that can effectively control saltwater intrusion. The main
goal of this study is to present a new saltwater intrusion control method that overcomes
most, if not all the limitations of previous methods and take into consideration different
parameters including: technical, economical and environmental aspects.
Chapter (1) Introduction
6
1.5 Proposed methodology to control saltwater intrusion
In this study a new methodology is proposed to study the control of saltwater intrusion.
This methodology aims to overcome the limitations of the previous methods. The proposed
methodology ADR (Abstraction, Desalination and Recharge) consists of three steps;
abstraction of brackish water from the saline zone, desalination of the abstracted brackish
water using reverse osmosis RO treatment process, and recharge of the treated water into
the aquifer (Figure 1.4). The main benefits of the ADR methodology are to return the
mixing zone into the original status and to reach a dynamic balance between fresh and
saline groundwater through two processes; abstraction of brackish groundwater to reduce
the volume of saline water and recharge of the treated brackish water to increase the
volume of fresh groundwater. The abstraction-recharge process helps to shift fresh
water/saltwater interface towards the sea and is considered an efficient method to retard
saltwater intrusion (Rastogi et al., 2004 and Mahesha, 1996c). The abstraction-recharge
process continues until a state of dynamic equilibrium is reached with respect to the salinity
distribution. A suitable quantity of treated brackish water is injected to the aquifer to
maintain the balance between fresh and saline groundwater and the excess of the treated
brackish water may be used for different purposes or stored in the aquifer.
Figure 1.4 The proposed methodology (ADR) to control saltwater intrusion
Saline groundwater
Sea
Fresh groundwater
RO
Chapter (1) Introduction
7
The second step of this methodology, desalination of brackish water using the RO treatment
process, aims to produce fresh water from the brackish water and use it for recharging the
aquifer to overcome the problem of scarcity of water in these areas. It is generally less
costly than other sources of freshwater for injection. For example, desalination of seawater
has many limitations such as; high cost, pollution (mainly carbon emission), and disposal of
the brine. Desalinating brackish water is an efficient alternative to seawater desalination,
because the salinity of brackish water is less than one-third of that of seawater. Therefore,
brackish water can be desalinated at a significantly lower cost. A simulation-optimization
model is developed to study different scenarios to control saltwater intrusion including;
abstraction of brackish water, recharge of fresh water and combination of abstraction and
recharge with RO desalination through the use of the proposed methodology ADR.
1.6 Effects of climate change and sea level rise on saltwater intrusion
One of the future challenges is climatic change and the related sea level rise. Climate
change is a result of natural and/or man-made activities. Due to climate change the
seawater levels might rise for several reasons including thermal expansion of oceans and
seas, melting of glaciers and ice caps and melting of Greenland and Antarctic ice sheets.
Sea level rise has many effects on coastal regions on the long term such as increase in
coastal erosion and seawater intrusion. Climate change has already caused changes in the
mean sea level during the last century by 10-20 mm/yr (IPCC, 1996). Future sea-level rise
due to climate change is expected to occur at a rate greatly exceeding that of the recent past.
It is expected the rise will be between 20–88 mm/yr (IPCC, 2001). The effects of climate
change and sea level rise on saltwater intrusion on the long term should be considered in
the modeling processes. The majority of the previous studies did not consider this point,
while few centimeters rise in sea level could have a great effect on saltwater intrusion. This
study considers the effect of increase in sea level due to climate change on the fresh
water/saltwater interface with the focus on determining the changes in the position of the
interface with time and identifying measures to control such movements. The predicted sea-
level rise data by IPCC (2001) will be incorporated into the developed simulation model to
investigate the new position of the fresh water/saltwater interface in coastal aquifers
considering different scenarios of pumping rates and sea level rise.
Chapter (1) Introduction
8
1.7 Objectives of the study
Saltwater intrusion problem is a serious problem that threatens the health and life style of
many people living in coastal areas. The problems of saltwater intrusion into groundwater
have become a considerable concern in many countries with coastal areas. Development of
a simulation-optimization model for the control of saltwater intrusion using different
management scenarios is the major aim of the present study. The model is based on the
integration of a genetic algorithm optimization technique and a coupled transient density-
dependent fluid flow and solute transport model, which has been developed for simulation
of seawater intrusion. The objectives of this work are:
To develop a coupled transient density-dependent FE model of fluid flow and solute
transport for computing solute transport in saturated/unsaturated soils. The model
includes, coupled water and air flow with energy transport and solute transport.
To apply the developed model to simulate water flow, air flow, heat transfer and
solute transport in saturated/unsaturated soils.
To apply the developed model to investigate the movement and the extent of
saltwater intrusion in coastal aquifers.
To develop a simulation-optimization model for the control of saltwater intrusion in
coastal aquifers using different management scenarios including; abstraction of
brackish water, recharge of fresh water and combination of abstraction and
recharge. The objectives of these management scenarios include minimizing the
total costs for construction and operation, minimizing salt concentrations in the
aquifer and determining the optimal depths, locations and abstraction/recharge rates
for the wells.
To apply a new methodology ADR, proposed in this research to control seawater
intrusion in real world coastal aquifers by abstracting brackish water, desalinating it
using small scale RO plant and recharging to the aquifer.
To apply the developed model to study the effects of likely climate change and sea
level rise on saltwater intrusion in coastal aquifers.
Chapter (1) Introduction
9
This study has touched on a serious real world problem which has caused considerable
concern in coastal regions in many countries throughout the world. The main contributions
of this study are (i) the development of a coupled transient variable-density FE model to
investigate seawater intrusion and study the effect of climate change and sea level rise on
coastal aquifers, (ii) development of management models to control seawater intrusion and
(iii) development of a simulation-optimization model to identify optimal arrangements for
the proposed control method (ADR). This work presents a new methodology ADR to
control saltwater intrusion in coastal aquifers, which is considered a first step in developing
a simulation-optimization model to control saltwater intrusion in coastal aquifers using
ADR. This methodology is considered to be cost-effective, capable of retarding
freshwater/saline water interface toward the sea, and a source of fresh water in coastal
regions.
1.8 Outline of the thesis
To achieve the study objectives, the thesis is organized in eight chapters. The main text of
each chapter is intentionally kept as short as possible in favour of easy reading and is
written to include only the fundamental concepts and the new ideas.
Chapter 2 presents background of saltwater intrusion investigation and control methods. A
review of saltwater intrusion investigation methods, mechanisms and numerical models
used for simulating saltwater is presented. The analyses of the numerical models that have
been used to control saltwater intrusion using different methods and advantages and
disadvantages of each method are presented. Also, a new methodology ADR is presented to
control saltwater intrusion.
In chapter 3, the mathematical formulations and governing equations which are used in the
development of this model are provided. These equations form a system of highly
nonlinear, coupled, partial differential equations. The governing equations can be divided
into two sets of equations, one set that describes the fluid flow and another set that
describes the solute transport. These equations can rarely be solved analytically for
situations of practical importance and so numerical solutions must be developed.
Chapter (1) Introduction
10
Chapter 4 presents an overview of the different mathematical models that have been used to
solve coupled fluid flow and solute transport problem focusing on finite element and finite
difference methods. The numerical solution of the governing differential equations of the
coupled fluid flow and solute transport model is presented and calibrated.
Chapter 5 presents validation of the developed model through numerous examples and case
studies to demonstrate the performance and accuracy of the developed numerical model.
Chapter 6 presents development and application of a simulation-optimization model to
control seawater intrusion in coastal aquifers using three management scenarios including;
abstraction of brackish water, recharge of fresh water and combination of abstraction and
recharge. The objectives of these management scenarios include minimizing the total costs
for construction and operation, minimizing salt concentrations in the aquifer and
determining the optimal depths, locations and abstraction/recharge rates for the wells.
In chapter 7, the causes of climate change and its effect on coastal areas are presented. In
particular, the effect of sea level rise on saltwater intrusion and the methods of control are
presented. The application of the developed model to study the effects of likely climate
change and sea level rise on saltwater intrusion in coastal aquifers is also presented.
Finally in chapter 8, the main conclusions of this work and recommendations for further
studies are presented.
Chapter (2) Saltwater intrusion investigation and control methods
11
Saltwater intrusion investigation and control methods
2.1 Introduction Groundwater is considered the main source of water supply in many coastal regions.
Population growth increases the water requirements leading to increase pumping from the
aquifers. Over-pumping is considered the main cause of saltwater intrusion; under normal
conditions freshwater flows into the sea, but over-abstraction may result in the inversion of
the groundwater flow from the sea towards the inland causing saltwater intrusion. Saltwater
intrusion can be defined as the inflow of seawater into an aquifer. It is common in coastal
aquifers where the aquifers are in hydraulic contact with the sea. Saltwater intrusion leads
to an increase in the saline water volume and a decrease in freshwater volume. Salinization
of groundwater is considered a special category of pollution because mixing a small
quantity of saltwater (2 to 3 percent) with groundwater makes freshwater unsuitable for use
and can result in abandonment of freshwater supply wells when salt concentration exceeds
drinking water standards. Therefore, saltwater intrusion should be prevented or at least
controlled to protect groundwater resources. The key to control saltwater intrusion is to
maintain a proper balance between water being pumped from the aquifer and water
recharged to the aquifer (Bear et al., 1999).
This chapter presents the background of saltwater intrusion investigation and control
methods. The chapter is divided into two main sections. A review of saltwater intrusion
investigation methods, mechanisms and numerical models used for simulating saltwater
intrusion in coastal aquifers is presented in the first section. The second section presents
and analyzes the numerical models that have been used to control saltwater intrusion using
different methods and the advantages and disadvantages of each method are highlighted.
Also, a new methodology ADR (Abstraction of brackish water, Desalination and Recharge
into the aquifer) is presented to control saltwater intrusion. The procedures and advantages
of the proposed methodology are presented and discussed.
Chapter (2) Saltwater intrusion investigation and control methods
12
2.2 Main causes of saltwater intrusion in coastal aquifers
Coastal aquifers generally lie within some of the most intensively exploited areas of the
world. About 70% of the world population lives in such areas. If current levels of
population growth and industrial development are not controlled in the near future, the
amount of groundwater use will increase dramatically, to the point that the control of
seawater intrusion becomes a major challenge to future water resources management
engineers. Saltwater intrusion is a serious problem in the coastal regions all over the world.
It may occur due to human activities and by natural events such as climate change and sea
level rise. The main causes of saltwater intrusion include (Bear et al., 1999):
1- Over-abstraction of the aquifers
2- Seasonal changes in natural groundwater flow
3- Tidal effects
4- Barometric pressure
5- Seismic waves
6- Dispersion
7- Climate change – global warming and associated sea level rise
Over-abstraction is considered one of the main causes of saltwater intrusion. Some of the
previous causes of saltwater intrusion are periodic (e.g. seasonal changes in natural
groundwater flow), some have short term implications (tidal effects and barometric
pressure) and others have long term implications (climate changes and artificial influences).
Salinization is one of the most widespread forms of groundwater contamination in the
world. Saltwater intrusion poses a major limitation to utilization of groundwater resources.
The intrusion of seawater should be controlled in order to protect groundwater resources
from depletion. In order to take measures to control and prevent seawater intrusion and to
get a clear understanding of the relationship between the consumption of groundwater and
seawater intrusion, in depth study either from the theoretical point of view or from the
numerical analysis and practical point of view is required.
Chapter (2) Saltwater intrusion investigation and control methods
13
2.3. Saltwater intrusion investigating methods
The analysis of salinization process in coastal regions requires information on both;
geological and hydrochemical characteristics of aquifers. The intrusion of saltwater in
coastal aquifers has been investigated by several methods including geophysical and
geochemical methods. A number of experimental studies have been presented and different
analytical and numerical models have been used to investigate saltwater intrusion in coastal
aquifers. These approaches attempted to ascertain the position of freshwater/saltwater
interface and predict changes in water levels and salinity. A brief description of the
methods that have been used is presented in the following sections.
2.3.1 Geophysical investigations
Saltwater intrusion can be distinguished by the movement of water with high total
dissolved solids (TDS) content into fresh water. The degree of salinity varies in the
transition zone between fresh and saline groundwater. The contour map of TDS is one
measure of salinity and the location of the contour of TDS=1000 mg/l marks the area under
the influence of seawater intrusion. The description of water type based on TDS content
suggested by USGS is shown in Table 2.1.
Table 2.1 The water type based on TDS content
TDS (ppm) or (mg/l) Description
< 1000 Fresh water
1000-3000 Slightly saline
3000-10000 Moderately saline
10000-35000 Very saline
>35000 Brine
Geophysical methods measure the spatial distribution of physical properties of the earth,
such as bulk conductivity of seismic velocity. Several geophysical methods can be used for
hydrogeologic investigations such as (Bear et al., 1999):
Chapter (2) Saltwater intrusion investigation and control methods
14
1- Surface geophysical methods
- Electrical methods, such as: DC resistivity, frequency domain electromagnetic
methods, airborne EM, loop-loop EM, time-domain electromagnetic sounding,
and very low frequency EM
- Seismic methods, such as: seismic refraction, and seismic reflection
- Ground penetrating radar
2- Borehole methods
- Electric logs
- Radiometric logs
- Integrated use of borehole logs
3- Integrated geophysical surveys
Numerous researchers used geophysical methods to investigate saltwater intrusion. Melloul
and Goldenberg (1997) analysed different methods of monitoring saltwater intrusion in
coastal aquifers using direct methods that include measurement of groundwater salinity
profiles and groundwater sampling of observation and pumping wells, and indirect
methods using geoelectromagnitics and the time domain electromagnetic method.
Mulrennan and Woodroffe (1998) examined the pattern and process of saltwater intrusion
into the coastal plains of the lower Mary River, northern Territory, Australia, due to the
effect of tides. Nowroozi et al. (1999) used the Schlumberger sounding resistivity methods
in the investigation of saltwater/freshwater interface in the eastern shore of Virginia.
Willert et al. (2001) used coastal aquifer test field (CAT-Field) to build a 3-D numerical
model for prediction of saltwater intrusion and water quality and quantity between
Bermerhaven and Cuxhaven, Germany, by DC-geoelectric measurements. Thomas et al.
(2001) presented a study to evaluate and determine the feasibility of partially replacing the
imported water with highly treated wastewater for injection to control saltwater intrusion.
Maimone (2001) developed a numerical model to simulate saltwater intrusion and used
time domain electromagnetic technique to locate the interface and confirm the accuracy of
the model results. Fitterman and Deszcz-Pan (2001) used airborne electromagnetic
technique for mapping saltwater intrusion in Everglades National Park, Florida.
Chapter (2) Saltwater intrusion investigation and control methods
15
Ekwurzel et al. (2001) carried out a hydrogeologic investigation using isotopes to
determine the source of salinization in the Souss-Massa Basin, Morocco. Tulipano and
Fidelibus (2002) presented the history of saltwater intrusion in the Salento Peninsula
Puglia, southern Italy during the last 30 years, and established a methodology to describe
and quantify salinization to Karstic aquifer subject to over exploitation. Rao and Rao
(2004) investigated the movement of saltwater/freshwater interface in Krishna delta, India.
They studied the control of saltwater intrusion using barriers of treated wastewater and
industrial effluents. Pina et al. (2004) investigated the island of Santiago to characterize
groundwater quality and identify the principal areas in risk of salt water intrusion in order
to develop mitigating measures to reduce saltwater intrusion. Lazar et al. (2004) measured
and characterized the fresh-saline interface fluctuation by sea tides by measurements of
electrical conductivity taken across the interface interval. Bocanegra and Massone (2004)
presented the results of analysis of coastal aquifers related to medium and large cities in
Latin America and the Caribbean. They also developed a model to prevent migration of
salts to protect the coastal aquifers. The study cases were applied to: Cuba, Mexico,
Colombia, Brazil and Argentina. Qurtobi et al. (2004) presented an investigation to
determine the mechanism of recharge to the aquifer and the region of salinity and its impact
on fresh groundwater resources in the Souss-Massa basin in the south west of Morocco
using isotopic techniques.
Levi et al. (2010) presented a geophysical study using combined high resolution electrical
resistance tomography and accurate time domain electromagnetic measurements to study
the relationships between seawater and groundwater via estuarine rivers at two sites along
the lower part of the Alexander river in Israel. The objective of the study was to delineate
the expected saline water intrusion from estuarine rivers into adjacent aquifers by means of
high resolution and accurate geoelectric/geoelectromagnetic methods. Using geophysical
techniques, Sherif et al. (2010) presented field investigations and measurements that
revealed the nature and extent of the seawater intrusion in the coastal aquifer of Wadi Ham,
UAE. They presented a 3D geological and true earth resistively modelling for the aquifer.
They used the results of the earth resistivity imaging surveys and chemical analyses of
collected water samples to obtain an empirical relationship between the inferred earth
resistivity and the amount of total dissolved solids.
Chapter (2) Saltwater intrusion investigation and control methods
16
2.3.2 Geochemical investigations
The distinction of different salinization mechanisms is crucial to the evolution of the origin,
pathways, rates and future salinization of coastal aquifers. Several geochemical criteria can
be used to identify the origin of salinity in coastal aquifers (Bear et al., 1999):
1- Cl-concentration: the Cl-concentration of 200 mg/l in groundwater is used as index of
saltwater intrusion.
2- Cl/Br ratios: the bromide ion can be considered as a good indicator of saltwater
intrusion. The concentration of bromide in fresh water is less than 0.01 mg/l.
3- Na/Cl ratios
4- Ca/Mg, Ca/(HCO3+SO4) ratios
5- O and H isotopes
6- Boron isotopes
A number of researchers used geochemical methods to investigate saltwater intrusion.
Zubari (1999) presented a hydrochemical study to identify the sources of aquifer
salinization and delineated their area of influence. Edet and Okereke (2001) examined the
extent of seawater intrusion using vertical electrical sounding and hydrochemical data into
shallow aquifers beneath the coastal plains of south eastern Nigeria. Gaye (2001) presented
the necessary roles of using integrated geochemical and isotopic methods to study
groundwater salinity problem in terms of its origin and prevention. Vallejos et al. (2004)
studied the hydrochemical data with conductivity and water temperature diagrams to assess
the process that causes changes in groundwater composition. Demirel (2004) used chemical
analysis and electrical conductivity measurements to show the history of saltwater intrusion
in time and space. Christensen et al. (2001) presented a study on both the geochemical and
physical aspects of saltwater intrusion and their interaction during an intrusion experiment
into a shallow aquifer in Shansehage, Denmark. Allen et al. (2001) presented a
multidisciplinary study involving hydrochemical sampling and borehole geophysics to
study the nature and occurrence of saline groundwater in the gulf islands, British Columbia,
Canada.
Chapter (2) Saltwater intrusion investigation and control methods
17
Sherif et al. (2005) conducted a 2-D earth resistivity survey in Wadi Ham, UAE in the area
between Fujairah and Kalba to delineate the seawater intrusion. They used Monitoring
wells to measure the horizontal and vertical variations in water salinity and thus to improve
the interpretation of earth resistivity imaging data. They used the results of vertical
electrical soundings and chemical analyses of collected water samples to obtain an
empirical relationship between the inferred earth resistivity and the amount of total
dissolved solids. This relationship was used along with the true resistivity sections resulting
from the inversion of 2D resistivity data to identify three zones of water-bearing formation
(fresh, brackish, and salt-water zones). EL Moujabber et al. (2006) used physico-chemical
analysis of water samples to investigate saltwater intrusion in the Choueifat-Rmeyle region,
Lebanon.
2.3.3 Experimental studies
Some researchers presented experimental studies to investigate saltwater intrusion. Koch
and Starke (2001) presented an experimental study using a Plexiglass for carrying out 2-D
tracer experiment of macrodispersion in density-dependent flow within a stochastic
realization of a heterogeneous media. Van Meir et al. (2004) developed an in situ laboratory
where an integrated set of tools were developed for characterising the saltwater intrusion
process in a hard rock aquifer with an appreciable level of heterogeneity. The final goal of
the geophysical and hydraulic investigations was to derive 3-D porosity and permeability
models as input for the analysis of specific saltwater intrusion experiments. Mukhopadhyay
et al. (2004) presented a laboratory study to investigate the effects or artificial recharge on
the aquifer material properties such as; porosity and permeability in group aquifer, Kuwait.
Motz (2004) used inactive cells to present the saltwater part of the aquifer and a no-flow
boundary along the interface. He performed a series of numerical experiments to
investigate how the saltwater-freshwater interface should be approximated in a groundwater
flow model to yield accurate values of hydraulic heads in the freshwater zone.
Chapter (2) Saltwater intrusion investigation and control methods
18
2.3.4 Mathematical models
Various analytical and numerical models have been developed to investigate saltwater
intrusion in coastal aquifers. Some of the important contributions are discussed here after a
comprehensive discussion in the following sections. Sherif et al. (1990) developed a finite
element model for flow and solute transport to investigate the effect of lowering
piezometric head due to excessive pumping on the saline water intrusion. Panday (1995)
provided perspectives on the analytical upconing solutions, and numerical solutions as
related to the physical system being modelled. Ghafouri and Parsa (1998) developed a finite
element model to simulate the hydrodynamics of flow and salt transport in well-mixed
estuaries in order to reduce the undesirable impacts of seawater intrusion. Tsutsumi et al.
(2001) developed a numerical model with a groundwater module to predict the movement
of freshwater/saltwater and the depth of freshwater/saltwater interface in coastal aquifers.
Teatini et al. (2001) developed a finite element model to simulate saltwater intrusion in
shallow confined aquifers. Qahman and Zhou (2001) applied SUTRA to simulate seawater
intrusion in the northern part of the Gaza strip, Palestine. Prieto et al. (2001) applied
SUTRA code to analyze the effects of seasonal variation in pumping and artificial and
natural recharge on the dynamics of saltwater intrusion in the Nitzamin, Israel.
Zhang (2002) constructed the characteristics finite element alternating-direction schemes
which can be divided into three continuous one-dimensional problems. This scheme was
used to analyze saltwater intrusion in three dimensions. Oude Essink and Schaars (2002)
developed a model to simulate density-dependent groundwater flow, heat and salinity
distribution, and seepage and salt load flux to the surface water system, using
MOCDENS3D code. The model was used to assess the effect of future development such
as climate changes, sea level rise, land subsidence as well as human activities on qualitative
and quantitative aspects of the groundwater system. Isikh and Karahanoglu (2004) used a
quasi three dimensional finite element model to simulate seawater intrusion into a coastal
aquifer below sea level mining operation. Vazquez et al. (2004) developed a numerical
model to investigate saltwater transport in Llobregat delta aquifer, Barcelona, Spain. Lee
(2004) investigated the effects of dewatering using SUTRA code to predict the movement
of seawater inland under various scenarios of recharge in the coastal plain.
Chapter (2) Saltwater intrusion investigation and control methods
19
2.4 History of saltwater intrusion modelling
Mathematical models help to understand the relevant processes that cause saltwater
intrusion in coastal aquifers (Bear et al., 1999). Over the years, a large number of analytical
and numerical models have been used to predict the location and movement of the
saltwater/freshwater interface. The numerical models can be categorized as sharp interface
models or diffusive (dispersive) interface models. The first attempt to model seawater
intrusion was made by Ghyben (1889) and Herzberg (1901). This simple model is known
as Ghyben- Herzberg model which assumes that saltwater and freshwater are immiscible
and separated by a sharp interface. Hubbert (1940) and Henry (1959) presented numerical
solutions of steady saltwater intrusion into coastal aquifer based on the assumption of sharp
interface.
This assumption treats the interface between saltwater and freshwater as sharp and well
defined interface. However, in practice the interface is not sharp and the saltwater merges
gradually with the freshwater by the process of mechanical dispersion. The width of the
dispersion zone depends on the characteristics of the aquifer and the movement of water
particles due to tides or the fluctuation due to recharge (Cooper, 1959). The assumption of
sharp interface can be applied in some field conditions to obtain a first approximation to the
freshwater pattern if the transition zone is relatively narrow otherwise the dispersion
process should be taken into account and the entrainment of saltwater by the moving
freshwater is necessary to describe the phenomena (Kohout, 1960). Over the years, several
mathematical models have been developed to predict the interface or transition zone
between freshwater and saltwater. Rifai et al. (1956) and Ogata (1959) indicated that the
dispersion coefficient is proportional to the velocity and it is much greater in the direction
of the velocity than in the lateral direction. Henry (1964) developed the first analytical
solution including the effect of dispersion in confined coastal aquifer under steady-state
conditions. Henry’s problem was later solved by Lee and Cheng (1974) in terms of stream
functions. Segol et al. (1975) developed the first transient solution based on a velocity-
dependant dispersion coefficient, using Galerkin finite element method. Numerous other
researchers, such as Pinder and Cooper (1970), Frind (1982) and Huyakorn et al. (1987),
Chapter (2) Saltwater intrusion investigation and control methods
20
employed numerical models in the simulation of saltwater intrusion problem using the
diffusive interface approach.
Extensive research is being carried out in many parts of the world with the objectives of
understanding the mechanism of seawater intrusion and improving the methods to control it
in order to protect groundwater resources in coastal aquifers. The developed numerical
models are based on different concepts and can be categorized as:
1- Sharp-interface or diffusive interface (density-dependent) models.
2- Saturated or unsaturated flow models.
3- Transient or steady flow models.
2.4.1 Sharp-interface or diffusive interface models
A number of mathematical models have been developed to simulate saltwater intrusion
based on sharp interface assumption. Recently, researchers have been focusing more on
diffusive interface (density-dependent) models which are more realistic.
2.4.1.1 Sharp-interface models
Modelling of seawater intrusion in coastal aquifers is not a new subject. The initial model
was developed independently by Ghyben (1988) and Herzberg (1901). This simple model
is known as Ghyben-Herzberg model and is based on the hydrostatic equilibrium between
fresh and saline water as shown in Figure 2.1, and is described by Ghyben-Herzberg
equation as;
fs hhfs
f )( (2.1)
where:
s : is the saltwater density
f : is the freshwater density
Chapter (2) Saltwater intrusion investigation and control methods
21
sh : is the depth of interface below mean sea level
fh : is the height of the potentiometric surface above the mean sea level
According to Ghyben-Herzberg equation, if the saltwater density is 1025 kg/m3 and
freshwater density is 1000 kg/m3, this gives fs hh 40 .
Figure 2.1 Hydrostatic equilibrium between fresh and saline water
Several mathematical and numerical models were developed to simulate saltwater intrusion
based on sharp interface assumption. Sbai et al. (1998) used the finite element method for
the prediction of saltwater intrusion under steady state and transient conditions based on a
sharp interface model assuming the fresh water and saltwater to be immiscible. Izuka and
Gingerich (1998) presented a method to calculate vertical head gradients using water head
measurements taken during drilling of a partially penetrating well. The gradient was then
used to estimate interface depth based of Ghyben-Herzberg relation. Naji et al. (1998) used
an optimization technique to determine the location of sharp interface between saltwater
and freshwater. The algorithm used was based on the combination of nonlinear
programming and h-adaptive boundary element method. Bower et al. (1999) presented an
analytical model for saltwater upconing in a leaky confined aquifer. The model assumed the
existence of a sharp interface between freshwater and saltwater.
Water Table
Saline groundwater
Fresh groundwater Sea
Ground surface
Sea level hf
hs Sharp interface
Chapter (2) Saltwater intrusion investigation and control methods
22
Fitzgerald et al. (2001) presented a series of inter-related codes that together can addressed
the issues related to effective management of coastal aquifer systems. This set of tools
included groundwater flow, solute transport, sharp interface and coupled flow and transport
modelling codes. All of these codes are 3-D and linked by common database (input/output)
files. Groundwater flow in the multi-aquifer system could be simulated and management
practices such as aquifer storage and retrieval, injection barrier wells, optimization of
pumping location and depths could be evaluated using these codes. Brunke and Schelkes
(2001) developed a program SIM-COAST to simulate the time-dependent behaviour in
coastal aquifers. In this approach the interface between freshwater and saltwater was
simplified to a line. Marin et al. (2001) used a quasi-three dimensional finite-difference
model, in which freshwater and saltwater flow were separated by a sharp interface, to
simulate groundwater flow in Karstic aquifer of northwestern Yucatan, Mexico.
Aharmouch and Larabi (2001) presented a numerical model, based on the sharp interface
assumption, to predict the location, shape and extent of the interface that occur in coastal
aquifer due to groundwater pumping. El Fleet and Baird (2001) developed a 2D-2 layer
sharp interface model to predict the impact of over-abstraction from the coastal aquifer on
saltwater intrusion, and applied the model to the coastal aquifer in Tripoli, Libya. Taylor et
al. (2001) presented a comparison between results of different codes applied to Henry’s
problem and analyzed the sensitivity of each code to changes in the parameters.
2.4.1.2 Diffusive interface models
Saltwater and fresh groundwater are separated by either a sharp interface or a transition
zone. Hydrodynamic dispersion and diffusion tend to mix the two fluids forming a
transition zone at the interface as shown in Figure 2.2. The transition zone depends on the
extent of seawater intrusion and aquifer properties. This transition zone is a result of
hydrodynamic dispersion of the dissolved matter and could be wide or narrow according to
the aquifer depth. Steady flow minimizes the transition zone thickness, whereas pumping,
recharge, tides and other excitations increase its thickness. The thickness of transition zones
could range from a few meters to several kilometres in over-pumped aquifers. The
thickness of a transition zone depends on (Todd (1974):
Chapter (2) Saltwater intrusion investigation and control methods
23
1- structure of the aquifer,
2- extraction from the aquifer,
3- variability of recharge,
4- tides, and
5- climate change.
Figure 2.2 Location of transition zone
A number of mathematical models have been developed to simulate saltwater intrusion
based on diffusive interface. Tasi and Kou (1998) developed a two dimensional finite
element code, considering density-dependant flow and transport. The model was used to
simulate saltwater intrusion and to evaluate the major effects of hydrological and geological
parameters, including hydraulic conductivity, hydraulic gradient on saltwater intrusion.
Benson et al. (1998) studied the effect of velocity gradient on solute transport. The study
showed that in case of saltwater intrusion, the velocity is so high and hence affects the
solute transport. Other numerical models assume approximate value of the velocity due to
its low value. Cheng et al. (1998) developed a 2-D finite element model for density-
dependant flow and solute transport through saturated and unsaturated soils. Sakr (1999)
presented a finite element model to simulate density-dependent solute transport. He used
the model to investigate the limitation of the sharp interface approach in coastal aquifers for
both steady and unsteady states.
Water Table
Saline groundwater
Fresh groundwater Sea
Ground surface
Sea level
Transition zone
Chapter (2) Saltwater intrusion investigation and control methods
24
Some researchers used computer codes to simulate saltwater intrusion. Paniconi et al.
(2001) used the CODESA-3D code that treats density-dependant variably saturated flow
and miscible solute transport to investigate the occurrence of seawater intrusion in the
Korba plain in northern Tunisia. Oude Essink (2001a,b) used MOCDENS3D for density-
dependant groundwater flow in three dimensions to simulate saltwater intrusion in the
island of Texel and also Noord-Holland Netherland. In addition, a relative sea level rise of
0.5 meter per century was considered. Payne et al. (2001) used SUTRA to predict the
effects of various pumping scenarios on groundwater supplies, in the context of
management of groundwater resources in coastal Georgia.
Bear et al. (2001) developed a 3-D finite element model that took into account the
development of a transition zone and variation of fluid density with time. The model was
used to analyze seawater intrusion induced by pumping wells in the upper section of the
coastal aquifer in Israel. Oswald et al. (2001) developed a 3-D model for variable density
flow to study the effect of several pumping scenarios on water quality in an island aquifer,
in China. Moe et al. (2001) presented a 3-D coupled density-dependent flow and transport
model for simulating seawater intrusion, migration of brine, and transport of brackish water
in the Gaza strip, Palestine. Lecca et al. (2001) presented a three-dimensional density-
dependent finite element model to simulate saltwater intrusion in the Sahel region of the
Atlantic coast, Morocco. Sabi et al. (2001) developed finite element model to predict the
interface in steady and transient conditions. The model was applied to the coastal aquifer of
Martil in Morocco to study the aquifer response to change in recharge and total rate of
pumping water and their effects on saltwater intrusion.
A numerical model to study density-dependent groundwater flow and solute transport in
unsaturated soil was developed by Jung et al. (2002). Gingerich and Voss (2002) expanded
a cross-sectional 2-D model developed by (Voss, 1984) to 3-D. The developed model
showed the effect of variable-density flow and solute transport on groundwater flow,
freshwater/saltwater zone and salinity distribution. Abarca et al. (2002) studied the effect of
aquifer bottom topography on seawater intrusion, using a three dimensional density-
dependant groundwater flow and transport model. Lecca et al. (2004) used a 3-D density-
Chapter (2) Saltwater intrusion investigation and control methods
25
dependant flow and contaminant transport code to improve the understanding of
groundwater degradation mechanism within the complex hydrodynamic system in the
coastal plain, Muravera, Italy.
Hamza (2004) presented a numerical model considering the dispersion zone between
freshwater and saltwater. The model was used to predict the effect of saltwater upconing on
the salinity of pumped water from a pumping well. He considered the effect of velocity-
dependent, hydrodynamic dispersion and density-dependant flow. The study was conducted
to determine the most important parameters that affect the quality of pumped water.
Stuyfzand et al. (2004) used salinization potential for water resources (SAPORE) index to
assess the use of brackish groundwater for drinking on the upconing of saline water. They
used reverse osmosis to treat brackish groundwater. Voudouris et al. (2004) described the
methods of the salinity distribution by seawater intrusion based on chemical analysis of
samples collected from the aquifer of Greece. Bakkar et al. (2004) developed a package for
modelling of regional seawater intrusion with MODFLOW, and used it to simulate the
evolution of the three-dimensional salinity distribution through the time considering the
effects of variation in density.
Lakfifi et al. (2004) developed a numerical model based on the SEWAT code to study
groundwater flow and saltwater intrusion in the coastal aquifer of Chaouia Morocco.
SEWAT code was used to simulate the spatial and temporal distribution of hydraulic head
and solute concentration. Doulgeris and Zissis (2005) developed a 3-D finite element model
for the simulation of saltwater intrusion in coastal confined aquifers by considering the
development of a transition zone between fresh water and seawater and thus density-
dependant flow and transport. Shalabey et al. (2006) presented a numerical model of
advective and dispersive saltwater transport below a partially penetrating pumping well.
The numerical solusion included a solution of the coupled flow and solute transport
equations. Canot et al. (2006) developed a 2-D numerical model for solution of transient
density driven flow in porous media.
Chapter (2) Saltwater intrusion investigation and control methods
26
2.4.2 Steady-state and transient flow models
Numerous models have been presented to study coupled fluid flow and solute transport in
porous media. Several researchers assumed steady-state solution for simplicity and avoided
the transient simulation because of high computational requirements.
Bixio et al. (1998) developed a fully coupled steady-state flow and solute transport finite
element model for the prediction of the maximum extent of saltwater intrusion in a vertical
section in the Venice aquifer, Italy. Sadeg and Karahannoglu (2001) developed a two-stage
finite element simulation model. In the first atage, an areal 2-D model was formulated to
perform a steady-state calibration for the physical parameters and boundary conditions of
the hydrostatic system. In the second stage, seawater intrusion was analysed using cross-
sectional finite element model.
Sbai et al. (1998) developed a finite element model for prediction of saltwater intrusion
under steady-state and transient flow conditions. Sakr (1999) developed a 2-D finite
element model to simulate density-dependent solute transport to investigate the limitation
of the sharp interface approach in coastal aquifers for both steady-state and unsteady state
conditions. Liu et al. (2001) presented a 2-D finite volume model to simulate saltwater
intrusion in coastal aquifers, the flow was considered transient and the variability of fluid
density was taken into account. Bear et al. (2001) developed a 3-D model using the finite
element framework. The model took into account the development of a transition zone and
variation of fluid density with time. Shalabey et al. (2006) developed a numerical model of
advective and dispersive saltwater transport below partially penetrating pumping wells.
They presented a solution of the coupled flow and solute transport equations that permits
estimation of salt concentration in the aquifer and water pumped from wells in time and
space.
Chapter (2) Saltwater intrusion investigation and control methods
27
2.4.3 Saturated and unsaturated flow models
Various models were developed to simulate solute transport in saturated soil. However, few
models were developed to simulate solute transport in unsaturated zone.
Teatini et al. (2001) developed a finite element model for saturated flow and transport to
simulate saltwater intrusion in shallow confined aquifers. Cau et al. (2002) developed a 3-D
model supported by GIS to investigate the causes of saltwater intrusion, using coupled flow
and solute transport in saturated porous media. The model was used to predict the
movement of the freshwater-saltwater mixing zone under different future scenarios.
Jung et al. (2002) presented a 3-D numerical model to investigate density-dependent
groundwater flow and solute transport in unsaturated soil. Cheng et al. (1998) developed a
2-D finite element model for density-dependant flow and solute transport through
saturated/unsaturated porous media to simulate saltwater intrusion. Paniconi et al. (2001)
developed a numerical model that treats density-dependant flow and miscible salt transport.
The developed model was used to investigate the occurrence of seawater intrusion in the
Korba plain in northern Tunisia. They also examined the effect and interplay between
pumping, artificial recharge, soil, aquifer properties and the unsaturated zone. A three
dimensional saturated-unsaturated finite element code, CODESA-3D was used in this
study. Voss (1984) employed a 2-D hybrid finite element model integrated with the finite
difference technique to simulate the flow of density-dependent fluid and solute transport in
saturated and unsaturated soil. Voss and Provost (2001) used the 3-D SUTRA code to
simulate variable-density flow and saltwater intrusion in coastal aquifers, which was
applied to study the intrusion of seawater in a number of real cases studies. Gingerich and
Voss (2002) expanded a cross-sectional 2-D model analysis (Voss, 1984) to 3-D using GIS.
This 3-D model produced the overall behaviour of groundwater flow and the salinity
distribution depending on variable-density flow and solute transport.
Chapter (2) Saltwater intrusion investigation and control methods
28
2.4.4 Recent computer codes to simulate saltwater intrusion
Over the years, a large number of models and computer codes have been developed and
used to study saltwater intrusion in coastal aquifers. These include analytical and numerical
models. Finite difference and finite element were widely used in this field. A number of
computer codes have been used to model coupled flow and solute transport and saltwater
intrusion in coastal aquifers. Summary of the common computer codes used to investigate
saltwater intrusion in coastal aquifers is presented below.
SUTRA
SUTRA is a 2-D and 3-D finite element code. SUTRA can simulate density-dependent
groundwater flow with energy transport or solute transport (Voss, 1984). This code has
been widely used to simulate variable-density groundwater flow. Gotovac et al. (2001) used
SUTRA to examine the effects of geologic heterogeneity upon the development of the
transition zone in coastal aquifer subject to seawater intrusion. The results indicated that
heterogeneity plays an important role in the displacement of the transition zone and the
variability of salt concentration. Gurdu et al. (2001) used SUTRA to simulate saltwater
intrusion in the Goksu delta at Silifke, Turkey. Narayan et al. (2002) used SUTRA to define
the current and potential extent of saltwater intrusion in the Burdekin delta aquifer under
various pumping and recharge conditions. A 2-D vertical cross-section model was
developed for the area, which accounted for groundwater pumping and artificial recharge
schemes. The results addressed the effects of seasonal variation in pumping rate and
artificial and natural recharge rates on the dynamics of saltwater intrusion. Alf et al. (2002)
presented a combination of SUTRA and hydrochemical investigations to investigate
saltwater intrusion in coastal aquifers. Ojeda et al. (2004) modelled the saline interface of
the Yucatan peninsula aquifer using SUTRA. The results showed that the interface position
is very sensitive to head changes.
SEWAT
SEWAT is a combination of MODFLOW and MT3DMS. It is designed to simulate 3-D
variable density groundwater flow and solute transport (Zheng and Wang 1999). This code
was used by a number of researchers to simulate different cases of saltwater intrusion. Rao
Chapter (2) Saltwater intrusion investigation and control methods
29
and Rao (2004) developed a simulation-optimization approach to control saltwater intrusion
through a series of extraction wells. SEAWAT was used to simulate the dynamics of
saltwater intrusion and the simulated annealing algorithm was used to solve the
optimization problem. Tiruneh and Motz (2004) investigated the effect of sea level rise on
the freshwater-saltwater interface, using SEWAT. Lakfifi et al. (2004) applied SEWAT to
study groundwater flow and saltwater intrusion in the coastal aquifer of Chaouia in
Morocco. Qahman and Larabi (2004) used SEAWAT code for simulating the spatial and
temporal evolution of hydraulic heads and solute concentrations of groundwater in the Gaza
aquifer (Palestine). The results showed that seawater intrusion would become more sever in
the aquifer if the current rates of groundwater pumping continued.
CODESA-3D
CODESA-3D is a 3-D finite element model that can simulate saltwater interface in
saturated and variably saturated porous media by solving the convective-dispersive
transport equation. Barrocu et al. (2004) used CODESA 3-D with GIS to study the
migration of contaminant and the impact of land management using different groundwater
abstraction schemes and artificial recharge in Sardinia, Italy. Paniconi et al. (2001) used
CODESA-3D to investigate the occurrence of seawater intrusion in the Korba plain in
northern Tunisia. They examined the effect and interplay between pumping, artificial
recharges, soil/aquifer properties and the unsaturated zone.
SWIFT
SWIFT is a 3-D code to simulate groundwater flow, heat transfer and brine and radio
nuclide transport in porous and fractured media (Ward and Benegar 1998 and Ward 1991).
Ma et al. (1997) used SWIFT to simulate saltwater upconing and its effect on the salinity of
pumped water.
MOCDENS3D
MOCDENS3D is a 3-D code to simulate density-dependant groundwater flow and solute
transport (Oude Essink, 1998). Oude Essink (2001c) used the code to model the movement
of freshwater, brackish and saline groundwater in coastal aquifers including advection and
dispersion.
Chapter (2) Saltwater intrusion investigation and control methods
30
SHARP
Barreto et al. (2001) used a finite difference model (SHARP) to simulate groundwater flow
in aquifers and established a relationship between the pumping flow, the location of the
wells, the number of the wells, the drop of freshwater heads, and the movement of
interface.
Quasi-three-dimensional finite element
Zhou et al. (2003) used a quasi-three-dimensional finite element to simulate the spatial and
temporal distribution of groundwater levels in the three-aquifer system. Mahesha (1996c)
used a quasi-three-dimensional finite element to examine the efficiency of battery of
injection wells in controlling saltwater intrusion.
SHEMAT
Scholze et al. (2002) applied SHEMAT to simulate saltwater intrusion in the coastal
aquifers of Metro Cebu.
Some other codes have been used to simulate saltwater intrusion in coastal aquifers (e.g.,
FEFLOW; HST3D; FAST-C (2D/3D) and MOCDENSE), (Beer et al., 1999).
Chapter (2) Saltwater intrusion investigation and control methods
31
2.5 Saltwater intrusion control methods
Considerable attention has been recently focused on models to study the control of
saltwater intrusion in order to protect groundwater resources. The control of saltwater
intrusion representing a continues challenge because groundwater provides about one-third
of the total freshwater consumption in the world especially in the coastal areas where 70 %
of the world population lives (Bear et al., 1999). The key to control saltwater intrusion is to
maintain a proper balance between water being pumped from the aquifer and water
recharged to the aquifer. Over the years, a number of methods have been used to control
saltwater intrusion in coastal aquifers. Not all the solutions are economically feasible
because they are considered long term solutions, and the time to reach the state of dynamic
equilibrium between freshwater and brackish water may takes tens of years (Bear et al.,
1999).
Different measures have been presented to control saltwater intrusion in coastal aquifers.
Todd (1974) presented various methods of preventing saltwater from contaminating
groundwater sources including:
1- reduction of pumping rates
2- relocation of pumping wells
3- use of subsurface barriers
4- natural recharge
5- artificial recharge
6- abstraction of saline water
7- combination techniques
Numerical models have been developed and used to help understanding the relevant
process that causes saltwater intrusion in coastal aquifers and identify suitable methods of
control. Extensive research has been carried out to investigate saltwater intrusion in coastal
aquifers. However, only limited amount of research has been directed to study the control
of saltwater intrusion. The following sections present the previous methods that have been
used to control saltwater intrusion in different locations. Also, the advantages and
disadvantages of each one are discussed (Abd-Elhamid and Javadi, 2008a).
Chapter (2) Saltwater intrusion investigation and control methods
32
2.5.1 Reduction of pumping rates
The natural balance between freshwater and saltwater in coastal aquifers is disturbed by
abstraction and other human activities that lower groundwater levels, reduce the amount of
fresh groundwater flowing to the sea, and ultimately cause saltwater to intrude coastal
aquifer. Increasing pumping rate is considered the main cause of saltwater intrusion along
the coasts. Other parameters that cause saltwater intrusion have smaller impact in
comparison with pumping. Population growth in coastal areas has increased water demand
which has in turn resulted in increase in abstraction from aquifers. The reduction of
pumping rates aims to achieve the sustainable yield and use other water resources to supply
adequate water demand. This can be achieved by a number of measures including;
1- increase in public awareness of the necessity of water to save water,
2- reduction of losses from the water transportation and distribution systems,
3- reduction of water requirement in irrigation by changing the crop pattern and using
new methods for irrigation such as drip irrigation, canal lining, etc,
4- recycling of water for industrial uses, after appropriate treatment,
5- reuse of treated waste water in cooling, irrigation and recharge of groundwater, and
6- desalination of seawater.
A number of models have been developed to control saltwater intrusion by reducing
pumping rates from the aquifer or using optimization models to optimize the abstraction
and control the intrusion of saline water. Scholze et al. (2002) used the SHEMAT code to
simulate saltwater intrusion in coastal aquifer of Metro Cebu. They applied different
scenarios of abstraction and decrease in water consumption, to protect groundwater
resources and avoid saltwater intrusion. Zhou et al. (2003) used a quasi-three-dimensional
finite element model to simulate the spatial and temporal distribution of groundwater
levels. The objective of the model was to maximize the total groundwater pumping from
the confined aquifer and control saltwater intrusion by relocating the wells. Amaziane et al.
(2004) coupled the boundary element method and a genetic algorithm for the optimization
of pumping rates to prevent saltwater intrusion.
Chapter (2) Saltwater intrusion investigation and control methods
33
Qahman and Larabi (2004) investigated the problem of extensive saltwater intrusion in
Gaza aquifer, Palestine using the SEWAT code. Different scenarios were considered to
predict the extension of saltwater intrusion with different pumping rate over the time.
Bhattacharjya and Datta (2004) used an artificial neural network as a simulator for flow and
solute transport in coastal aquifer to predict salt concentrations of the pumped water. A
linked simulation-optimization model was formulated to link the trained ANN with a GA
based optimization model to solve saltwater management problems. The objective of the
management model was to maximize the permissible optimal abstraction of groundwater
from the coastal aquifer for beneficial use, while maintaining salt concentration in the
pumped water under specific permissible limits.
Advantages: Reduction of abstraction rates and use of other water resources help to
increase the volume of freshwater which retards the intrusion of saltwater.
Disadvantages: Control of saltwater intrusion using this method has some limitations. The
control of abstraction rates cannot be fully achieved especially from private stakeholders.
Also the alternatives proposed for facing the shortage of water and the increased demands
may be very costly, especially when freshwater is not available and requires transportation.
Desalination of seawater as an alternative has also a lot of disadvantages; it is still very
expensive, is a source of pollution and requires a large area of land. The treated waste water
is also costly. Furthermore, this can only be a temporary solution because it does not
prevent saltwater intrusion but it attempts to reduce it, but with the population growth and
increasing demands the problem will not be controlled (Abd-Elhamid and Javadi, 2008a).
2.5.2 Relocation of pumping wells
Changing the location of pumping wells by moving the wells to more inland positions aims
to raise the groundwater level and maintain the groundwater storage. This is because in the
inland direction the thickness of the freshwater lens increases and the risk of upconing of
saltwater decreases accordingly.
Chapter (2) Saltwater intrusion investigation and control methods
34
Ofelia et al. (2004) used a mathematical model to identify saltwater volume interred the
fresh groundwater in Santa Fe, Argentina. The study examined a management model to
protect the fresh groundwater system. Several pumping wells were removed from services
and a new pimping field was designed and implemented. Hong et al. (2004) developed an
optimal pumping model to evaluate optimal groundwater withdrawal and the optimal
location of pumping wells in steady-state condition while minimizing adverse effects such
as water quality in the pumping well, drawdown, saltwater intrusion and upconing. The
study involved experimental verification of the optimal pumping model to develop
sustainable water resources in the coastal areas.
Ojeda et al. (2004) modeled the saline interface of the Yucatan peninsula aquifer using
SUTRA assuming an equivalent porous medium with laminar flow and two dimensional
flow. The models showed that the interface position is very sensitive to head changes and a
simple groundwater abstraction scheme was presented which was based on the well depth
and the distance from the coast. Maimone and Fitzgerald (2001) applied three dimensional
groundwater flow models, dual phase sharp interface intrusion model, radial upconing
model, and single phase contaminant transport model to develop a coastal aquifer
management plan. Two techniques were used; one based on developing new well locations
inland from the coast and the other one on use of RO treatment for desalinating brackish
water and useing it for domestic purposes. Sherif et al. (2001) used two models to simulate
the problem of saltwater intrusion in the Nile Delta aquifer in Egypt in the vertical and
horizontal directions. The two models were 2D-FED and SUTRA. The 2D-FED model was
employed to simulate the current condition and predict the effect of the seawater level rise
in the Mediterranean Sea under the condition of global warming. SUTRA was used to
define the best location of additional groundwater pumping wells from the Nile Delta
aquifer and to assess the effect of various pumping scenarios on the intrusion process.
Advantages: Moving pumping wells further inland helps to decrease the occurrence of
upconing of saltwater (Abd-Elhamid and Javadi, 2008a).
Disadvantages: Control of saltwater intrusion by this method is costly and may face some
obstructions such as, buildings or the size of the aquifer may not allow such movment. It is
also a temporary solution and does not prevent the intrusion of saline water into the aquifer.
Chapter (2) Saltwater intrusion investigation and control methods
35
2.5.3 Subsurface barriers
This method involves establishment of a subsurface barrier to reduce the permeability of
the aquifer to prevent the inflow of seawater into the basin. Construction of barriers could
be achieved using sheet piling, cement grout, or chemical grout. Figure 2.3 shows a sketch
of a subsurface barrier to control saltwater intrusion.
Figure 2.3 Sketch of a subsurface barrier
Barsi (2001) developed two methods for optimal design of subsurface barriers to control
seawater intrusion through the development of implicit and explicit simulation-optimization
models. The main objective was to find the optimal design of subsurface barrier to
minimize the total construction cost through the selection of the width and location of the
barrier. Harne et al. (2006) presented a 2-D sub-surface transport model of saltwater
considering the soil to be homogenous and isotropic under the influence of constant
seepage velocity. They used the finite difference method to solve the transport equation.
The model examined the efficiency of subsurface barrier to control saltwater intrusion.
James et al. (2001) developed a process for selectively plugging permeable strata with
microbial biofilm. These biofilm barriers can aid the prevention of saltwater intrusion by
reducing the subsurface hydraulic conductivity. The main advantage offered by the biofilm
Saline groundwater
Sea
Subsurface barrier
Fresh groundwater
Chapter (2) Saltwater intrusion investigation and control methods
36
barrier technology is that biofilm barrier construction can be achieved without excavation
and therefore will be economic.
Advantages: Using subsurface barriers helps to reduce the intrusion of saline water.
Disadvantages: Control of saltwater intrusion using subsurface barriers could be costly in
terms of construction, operation, maintenance and monitoring. It is also not efficient for
deep aquifers (Abd-Elhamid and Javadi, 2008a).
2.5.4 Natural recharge
This method aims to feed aquifers with additional surface water by constructing dams and
weirs to prevent the runoff from flowing to the sea. This method can also be used for flood
protection. The retained water infiltrates into the soil and increases the volume of
groundwater storage. Figure 2.4 shows a sketch of the natural recharge process.
Figure 2.4 Sketch of the natural recharge process
This method could be efficient for unconfined aquifers but it could take a long time to
recharge the aquifer depending on its properties. Ru et al. (2001) used a quasi three-
dimensional model to simulate the movement of the interface between seawater and
freshwater and evaluate the effect of constructing a dam to protect groundwater resources.
Recharge
Saline groundwater
Fresh groundwater Sea
Dam
Chapter (2) Saltwater intrusion investigation and control methods
37
The function of the subsurface dam was to collect the rainfall water and recharge the
aquifer to increase the groundwater storage and retard saltwater intrusion. Bajjali (2005)
applied geostatistical techniques of GPI, IDW with GIS, to examine the effect of recharge
dam on groundwater quality. The infiltrated water below the dam increases the aquifer
storage of freshwater and pushes the saline water toward the sea.
Advantages: This method helps to prevent the runoff to flow directly to the sea and uses it
to increase the groundwater storage in the aquifer and prevent the intrusion of saline water.
Disadvantages: Natural recharge depends on the soil properties and requires high
permeability soil. Depending on the soil permeability, the recharge process could take a
long time. The cost of construction of dams and weirs and their maintenance is very high.
This solution is unsuitable for confined and deep aquifers.
2.5.5 Artificial recharge
Artificial recharge aims to increase the groundwater levels, using surface spread for
unconfined aquifers and recharge wells for confined aquifers. The potential sources of
water for injection may be from surface water, pumped groundwater, treated wastewater,
desalinated seawater, or desalted brackish water. Surface water can be taken from rivers or
canals through pipelines. Figure 2.5 shows a sketch of a recharge well.
A number of researchers used this method to control saltwater intrusion in coastal aquifers.
Narayan et al. (2006) used SUTRA to define the current and potential extent of saltwater
intrusion in the Burdekin Delta aquifer under various pumping and recharge conditions. A
2-D vertical cross-section model was developed for the area, which accounted for
groundwater pumping and artificial recharge schemes. The results addressed the effects of
seasonal variation in pumping rate and artificial and natural recharge rates on the dynamic
of saltwater intrusion. The simulation was carried out for a range of recharge, pumping
rates and hydraulic conductivity. Papadopoulou et al. (2005) developed a 3-D finite
element-finite difference groundwater flow simulation model. The extension of the
saltwater front along the coastal zone was calculated only hydraulically because his model
did not consider diffusion due to different densities of saltwater and freshwater. Artificial
Chapter (2) Saltwater intrusion investigation and control methods
38
recharge was presented using different scenarios including different well locations and
different injection rates. The source of water used for recharge was the effluent of the waste
water treatment plant of the industrial zone after tertiary treatment.
Figure 2.5 Sketch of recharge and abstraction wells
Kashef (1976) studied the effect of recharge wells of different patterns on the degree of
saltwater retardation in confined coastal aquifers. The batteries (recharge wells) were
located at various distances parallel to the shoreline; each battery consisting of equally
spaced recharge wells of a certain number. It was found that, the optimum location of the
batteries should lie between 0.7L and L, where L is initial length of the intruded saltwater
wedge. Mahesha (1996a) studied the effect of battery of injection wells on seawater
intrusion in confined coastal aquifers. He used a quasi three-dimensional areal finite
element model considering a sharp interface. He studied various conditions by changing the
well spacing and intensity and duration of fresh water injection. He concluded that, spacing
between wells, injection rate and duration of injection control the repulsion of the saline
wedge.
Mahesha (1996b) presented steady state solutions for the movement of the fresh
water/seawater interface due to a series of injection wells in a confined aquifer using a
sharp interface finite element model. The model was used to perform parametric studies on
Abstraction well
Saline groundwater
Sea
Recharge well
Fresh groundwater
Chapter (2) Saltwater intrusion investigation and control methods
39
the effect of location of the series of injection wells, spacing of the wells and fresh water
injection rate on the seawater intrusion. It was found that reduction of seawater intrusion
(of up to 60-90 %) could be achieved through proper selection of injection rate and spacing
between the wells. Also, he studied the effect of double series of injection wells and
compared with the single series. He found that a double series performs slightly better than
single series. Also, it was found that the staggered system of wells in the series is slightly
better than the straight well system for long spacing. Lopez-Ramirez et al. (2006)
presented a study to define the optimum conditions for physiochemical pre-treatment of
secondary effluents for successful reverse osmosis operation for groundwater recharge.
They also examined the suitability of treated waste water for groundwater recharge and
concluded that high quality reclaimed wastewater can be safely used in groundwater
recharge. Paniconi et al. (2001) developed a numerical model that treats density-dependant
variably saturated flow and miscible salt transport to investigate the occurrence of seawater
intrusion in the Korba plain in northern Tunisia. They examined the effect and interplay
between pumping, artificial recharge, soil/aquifer properties and the unsaturated zone.
A quasi three-dimensional groundwater flow model was developed using MODFLOW to
simulate groundwater basin flow characteristics in response to pumping and recharge
scenarios by Johnson et al. (2001). Three major steps were taken to control saltwater
intrusion; construction of saltwater barriers, basin adjudication to set a limit on
groundwater extraction and artificial recharge. Liles et al. (2001) developed a groundwater
model using MODFLOW with the objective of forecasting injection quantities and well
locations needed to prevent saltwater intrusion. They used highly treated wastewater in
injection into Orange County aquifers to prevent saltwater intrusion. Tompson et al. (2001)
applied integrated isotopic characterization and numerical modelling techniques to assess
the migration of reclaimed water that was used for artificial recharge in the coastal aquifer
underlying Orange County, California to characterize the age of groundwater recharged in
order to determine the distance between the recharge and production wells. It was shown
that the recharged reclaimed wastewater must reside below the surface for at least one year
before it could be produced for drinking water. This would present a technical challenge to
the commercial and public sector hydrologic community.
Chapter (2) Saltwater intrusion investigation and control methods
40
Paniconi et al. (2001) used CODESA-3D to investigate the occurrence of seawater
intrusion in eastern Cap-Bon, Tunisia. The model examined the interplay between pumping
regime and recharge scenarios and its effect on controlling the saline water intrusion.
Fitzgerald et al. (2001) presented a series of inter-related codes that together addressed the
issues related to effective management of coastal aquifer systems. This set of tools included
groundwater flow, solute transport and coupled flow and transport modeling codes. All of
these codes were 3-D and linked by common database (input/output) files. Groundwater
flow in a multi-aquifer system was simulated. Management practices such as aquifer
storage and retrieval, injection barrier wells, optimization of pumping location and depths
were evaluated.
Mukhopadhyay (2004) presented a laboratory study to investigate the effects or artificial
recharge on the aquifer properties; porosity and hydraulic permeability. Recharged water
may be groundwater-desalinated seawater mixed with 10% - treated waste water. Barrocu
et al. (2004) implemented the CODESA-3D code with GIS to study the migration of
contaminant and simulate the impact of land management using different groundwater
abstraction schemes and artificial recharge in Sardinia, Italy. Vandenbohede et al. (2008)
presented a simulation model to study sustainable water management using artificial
recharge of fresh water in the dunes of the western Belgian coastal plain by two recharge
ponds. The recharged water was produced from secondary treated waste water effluent by
the combination of ultra filtration and reverse osmosis.
Advantages: Artificial recharge helps to increase the groundwater storage in the aquifer and
prevent the intrusion of saline water.
Disadvantages: The artificial recharge has been discussed in many papers as a solution to
control saltwater intrusion. However, some of these models ignored the source of fresh
water especially in the areas that suffer from scarcity of water. On the other hand the cost of
fresh water in cases where it is not available in these areas and transported from other
places has not been taken into consideration. Using desalinated seawater is costly and could
be a source of pollution. The treated wastewater was used in many areas for recharge to
control saltwater intrusion. This technique is often costly and ineffective in the areas where
excessive groundwater pumping occurs (Narayan et al. 2002 and Narayan et al. 2006).
Chapter (2) Saltwater intrusion investigation and control methods
41
Treated waste water affects the aquifer properties and requires at least one year before
abstraction from the aquifer. At present there is a growing opposition against artificial
recharge by infiltration at the land surface, because it occupies a large area.
2.5.6 Abstraction of saline water
This method of control aims to reduce the volume of saltwater by extracting brackish water
from the aquifer and returning to the sea. Figure 2.5 shows a sketch of an abstraction well.
Sherif and Hamza (2001) developed a finite element model (2D-FED) considering
dispersion and variable density flow to simulate the effect of pumping brackish water from
the transition zone in order to restore balance between freshwater and saline water in
coastal aquifer. The disposal of abstracted saline water was a problem. They proposed
different solutions to use abstracted saline water for some purposes such as; cooling,
desalting, injection into deep wells, irrigation, or disposal to the sea. Johnson and Spery
(2001) presented different methods to control saltwater intrusion in different states in the
USA. In California they extracted saline water and desalinated it using RO treatment
process. The treated water was blended with untreated groundwater to produce water
suitable for domestic delivery. In Los Angeles injection wells were used to protect the
coastal aquifers from saltwater intrusion. Potable and highly treated waste water was
injected into the wells.
Maimone and Fitzgerald (2001) applied three dimensional groundwater flow models, dual
phase sharp interface intrusion model, radial upconing model and single phase contaminant
transport model to develop a coastal aquifer management plan. Two techniques were used;
development of new well locations further inland and use of RO treatment for desalinating
brackish water and using it for domestic purposes. Sherif and Kacimov (2008) suggested
pumping brackish water, encountered between the freshwater and saline water bodies, to
reduce the extension of seawater intrusion. They used SUTRA to examine different
pumping scenarios in the vertical view and the equiconcentration lines and velocity vectors
were identified for the different cases. They concluded that seawater intrusion problems
could be controlled through proper pumping of saline groundwater from the coastal zone.
Chapter (2) Saltwater intrusion investigation and control methods
42
Kacimov et al. (2009) investigated seawater intrusion in coastal unconfined aquifers of
Oman, experimentally, analytically and numerically. Water table elevation, capillary fringe,
moisture distribution, EC were observed and measured in a pilot site Al-Hail, Oman.
Laboratory measurements of conductivity and capillary rise were conducted in repacked
columns in the laboratory. Evaporation from a shallow horizontal water table to a dry soil
surface was modelled by HYDRUS2D. An analytical Dupuit-Forchheimer model was
developed for the planar part of the catchment with explicit expressions for the water table,
sharp interface and stored volume of fresh water. SUTRA code is used to study a variable
density flow in a leaky aquifer with line sinks modelling fresh water withdrawal and
evaporation. Both analytical and numerical models proved that pumping of saline
groundwater from coastal aquifers would mitigate the migration of seawater into the aquifer
and would contribute to the enhancement of the groundwater quality that is consistent with
the findings of Sherif and Hamza (2001).
Advantages: Abstraction of saline water decreases the volume of saline water in the aquifer
and protects pumping wells from upconing.
Disadvantages: The main problem in abstraction of brackish water is the disposal of the
saline water. Many researchers have attempted to solve this problem but it is still a subject
of current research. Brackish water is suitable only for certain types of crops and using it in
cooling may cause corrosion to the systems. The disposal of brackish water into the sea
could affect the marine life in these areas, fishing and tourism activities. Another problem
is that increasing abstraction of saline water could, in some cases, increase the intrusion of
saltwater (Abd-Elhamid and Javadi, 2008a).
2.5.7 Combination techniques
Combination between two or more of the above methods can help to eliminate the
disadvantages of these methods and give better control of saltwater intrusion. Zhou et al.
(2003) and Hong et al. (2004) used a combination of reduction in pumping rates and
relocation of pumping wells to control saltwater intrusion. Maimone and Fitzgerald (2001)
used development of new well locations further inland and use of RO treatment for
desalinating brackish water and using it for domestic consumption to reduce the abstraction
Chapter (2) Saltwater intrusion investigation and control methods
43
from the aquifer. Narayan et al. (2006), Paniconi et al. (2001) and Barrocu et al. (2004)
used reduction of pumping rates and recharge of freshwater to the aquifer. Johnson et al.
(2001) used a combination of three techniques: subsurface barriers, reduction of abstraction
rates and recharge of freshwater. Fitzgerald et al. (2001) used change of well locations,
reduction of pumping rates and recharge of freshwater.
Combination of injection of freshwater and extraction of saline water can reduce the
volume of saltwater and increase the volume of freshwater. Only few models have been
developed for this technique. Georgpoulou et al. (2001) developed a decision aid tool to
investigate the feasibility and applicability of a strategy involving the use of desalinated
brackish groundwater, pumped from the aquifers, coupled with recharge of aquifers by
reclaimed wastewater to control saltwater intrusion.
Mahesha (1996c) studied the control of seawater intrusion by a series of abstraction wells
for saline water alone and also in combination with freshwater injection wells in a confined
aquifer using a vertically integrated two dimensional sharp interface model under steady
state conditions. It was found that the combination of injection wells with extraction wells
produced excellent results to the individual cases for larger well spacing and small rates of
injection. Rastogi et al. (2004) developed a two-dimensional steady state numerical model
to study seawater intrusion problem involving hydrodynamic dispersion in synthetic multi-
layered confined coastal aquifer. The model was used to investigate the efficiency of
seawater control measures involving two scenarios; (i) freshwater recharge, and (ii)
combination of freshwater recharge and saltwater discharge wells. The study found that
depth, location and head of recharge wells are important parameters that can control
saltwater intrusion. The model confirmed that combined recharge and discharge system is
more effective in controlling seawater intrusion.
Chapter (2) Saltwater intrusion investigation and control methods
44
2.6 A new methodology to control saltwater intrusion (ADR)
This work presents a new methodology ADR (abstraction, desalination and recharge) to
control saltwater intrusion in coastal aquifers. This method aims to overcome all or at least
most of the limitations of the previous models. This methodology can help to control the
saltwater intrusion in coastal aquifers considering different parameters such as; technical,
economical and environmental aspects. Figure 2.6 shows a sketch of the new methodology
(ADR) (Abd-Elhamid and Javadi, 2008a).
The proposed methodology ADR consists of three steps:
Abstraction: Abstraction of brackish water from the saline zone using abstraction wells.
Desalination: Desalination of the abstracted brackish water using small-scale RO plant.
Recharge: Recharge of the treated water using recharge wells into the aquifers.
Figure 2.6 Sketch of the new methodology (ADR)
Saline groundwater
Sea
Fresh groundwater
DR
LR LA
DA
QA QR RO
Chapter (2) Saltwater intrusion investigation and control methods
45
The proposed methodology considers:
1- the relationship between abstraction and recharge rates; QA and QR ,
2- the locations of abstraction and recharge wells; LA and LR ,
3- the depths of abstraction and recharge wells; DA and DR ,
4- the aquifer properties,
5- construction and operation costs,
6- environmental impacts, and
7- climate changes, sea level rise and their effects on saltwater intrusion.
2.6.1 The benefits of ADR methodology
This method is a combination of two methods; abstraction of saline water and recharge of
fresh water in addition to desalination of abstracted water and treatment to be ready for
recharge or domestic use. It combines the advantages of these three steps as:
The first step: Abstraction of brackish water helps to reduce the saline water volume in the
aquifer and reduce the intrusion of saltwater.
The second step: Desalination of abstracted brackish water using RO treatment process
aims to produce fresh water from brackish water for recharge. This step is
very important to produce freshwater in the areas where freshwater is
scarce. The advantages of RO as a desalination method are discussed in
details in the next section.
The third step: Recharge of treated water aims to increase the fresh groundwater volume
to prevent the intrusion of saltwater.
The combination of abstraction and recharge techniques is considered one of the most
efficient methods to control saltwater intrusion. It increases the volume of fresh
groundwater and decreases the volume of saltwater. This method is capable of retarding
saltwater intrusion. It also has lower energy consumption, lower cost and lower
environmental impact. ADR can be used to increase the water resources in coastal regions
by increasing the abstraction of saline water and desalination. The excess reclaimed water
can be used directly for different purposes or injected into the aquifer to increase the
groundwater storage (Abd-Elhamid and Javadi, 2008a).
Chapter (2) Saltwater intrusion investigation and control methods
46
2.6.2 Desalination of brackish groundwater using RO
Desalination of seawater and brackish groundwater is considered a non-conventional
method to produce freshwater in coastal areas. Desalination of brackish groundwater could
increase the water supply in the region since it is technologically feasible and usually will
not have much negative environmental impact. Brackish groundwater with an amount of
TDS between 1,000 and 10,000 mg/l, can be desalinated at relatively low cost (compared
with seawater which contains 35000 mg/l of TDS). Desalination is a process of removal of
dissolved minerals (mainly salts) from either seawater or brackish groundwater to produce
fresh water of potable water quality. The produced water is divided into two streams, one
for potable water with a low TDS and the second for brine which includes high percentage
of dissolved salts. Despite the evolution of desalination technologies, desalination has some
major problems such as; high cost, environmental effects, carbon emission, land use for
plants and noise, and disposal of brine waste which contain high salt concentration.
Discharge of brine waste with high salt concentration and high temperature into the
environment has many environmental impacts as it affects marine habitats at shores,
tourism industry, fishing and fisheries industry, changes in the sea currents, temperature,
salinity and density and increased salinity leading to increasing the salinity of groundwater.
The desalination can be accomplished by different techniques that can be classified under
three main processes; thermal processes, membrane processes and other processes (Ismail,
2003). Thermal process consists of applying heat to the source of water to change the water
phase to vapour and solid, separate salts and collect the condensed water vapour to produce
pure water. A number of thermal methods can be used for desalination such as: Multistage
Flash distillation, Multiple Effects Distillation and Vapour Compression. Membrane
processes aim to separate the components with a membrane and without phase change, as
they move in response to an externally applied pressure. Saline solution is pressurized to a
point greater than the osmotic pressure of the solution, to separate water from salts. Two
main processes of this type are used for desalination: Reverse Osmosis and Electro
Dialysis. Other new desalinating processes are also used based on different techniques such
as: ion exchange, membrane distillation, freezing separation, solar distillation, nuclear
desalination.
Chapter (2) Saltwater intrusion investigation and control methods
47
The choice of a desalination method is not an easy one. The desalination method can be
selected based on three main aspects; technical aspects, environmental aspects and
economical aspects (Bakheet, 2006). Power costs in water desalination may account for 30
to 60 % of the operational costs, or about 30% to 50% of the total cost of the produced
water depending on the type of energy used. So, if the power costs increase, there is a direct
impact on the cost of the desalinated water. The challenge in desalination process is the
reduction of energy used to reduce the total cost of the produced water. Several methods
can be used for desalinating brackish groundwater but RO has become more attractive
because it consumes less energy so it is cheaper and has less carbon emission, requires
simple equipments, does not need to be linked to power plant, requires small area of land
and produces less noise. In the last decades there has been a rapid progress in the RO
desalination technology aiming to decrease the power consumption. A sharp reduction in
the cost has occurred; the average price of desalination is one-tenth of what it was twenty
years earlier, dropping dramatically from ($5.50/m3) in 1970 to ($0.55/m3) in 1999,
including interest, capital recovery and operation and maintenance (Ismail, 2003).
Reverse osmosis is currently one of the fastest-growing techniques in water desalination.
RO is a widely used method for desalinating water by using a pressure driven technique. It
uses dynamic pressure to overcome the osmotic pressure of the salt solution. Water is
forced to flow through small pores under high pressure through semi-impermeable
membranes while salt is rejected. The pressure difference must be high enough to overcome
the natural tendency of water to move from the low salt concentration side of a membrane
to the high concentration side as defined by osmotic pressure. The major energy
consumption comes from creating pressure. Small pores require high pressure and
consequently more energy. The quality of produced water depends on efficiency of
membranes, pressure and degree of salinity. Membranes are sensitive to pH, oxidizers, a
wide range of organics, algae, and bacteria and of course particulates and other foulants.
Therefore, pretreatment of the feed water is important in order to prevent membrane
contamination and fouling where large particles are filtered. Pretreatment can offer a
significant impact on the cost reduction of RO. On the other hand, RO post-treatment
includes removing dissolved gasses (CO2), and stabilizing the pH via the addition of Ca or
Na salts. Figure 2.7 shows a schematic sketch layout of a RO plant (Bakheet, 2006).
Chapter (2) Saltwater intrusion investigation and control methods
48
Figure 2.7 Schematic sketch layout of RO desalination plant
Various models were developed to study the technical and economical feasibility of
desalting brackish groundwater using RO. Al-Hadidi (1999) presented an economic
feasibility study to desalination of brackish groundwater using RO and its use as a non-
convictional water resource for integrated water resources management plan. The problem
of the study was the use and disposal of brine waste. Georgpoulou et al. (2001) developed a
decision aid tool for the investigation of feasibility and applicability of an alternative water
supply strategy involving the use of desalinated brackish groundwater pumped from the
aquifers coupled with aquifer recharge by reclaimed wastewater. Abderahman (2001)
described the process of desalting brackish water and reuse after blending with freshwater.
Shahalam et al. (2002) developed a combining unit process for pre-treatment of RO to
improve the quality of feed water at a minimal cost and with minimal effects on succeeding
the process.
Ramirez et al. (2003) used physical-chemical pre-treatment for the reverse osmosis unit for
reclamation of secondary effluent. The produced water is suitable for injection into a
groundwater aquifer to counteract saltwater intrusion. Al-azzan et al. (2003) used RO brine
staging for desalination to increase the recovery rate to 97 % in Kuwait. El Sheikh et al.
(2003) presented a technical and economical study to the cost and power consumption of
reverse osmosis plants in Gaza strip. RO technology can substitute the deficit in domestic
Low pressure pump
Produced water to recharge well
Membranes
High pressure pump
Brine waste Feed water from abstraction well
Pre-treatment tank Post treatment
tank
Chapter (2) Saltwater intrusion investigation and control methods
49
water requirements with comparatively acceptable and affordable cost. Purnama et al.
(2003) studied the impact of brine disposal operations on coastal and marine environment.
These effects can be avoided by extending the outfalls further offshore to the sea.
Afonso et al. (2004) presented a study to assess the technical-economical feasibility of
brackish groundwater treatment by RO for the production of potable water from brackish
water in Mediterranean countries. The study proved that RO can produce potable water
with low operating cost, and recovery rate of about 75%. Jaber and Ahmed (2004)
presented technical and economical aspects of RO brackish water desalination plants in
Gaza strip and estimated the quantities of desalinated brackish water over the planning
period. Lopez-Ramirez et al. (2006) presented a study to determine the optimum conditions
for physiochemical pre-treatment of secondary effluents for successful reverse osmosis
operation for groundwater recharge. They concluded that high quality reclaimed
wastewater can be used safely in groundwater recharge.
According to the literature, desalination of brackish groundwater has been successfully
used as non-conventional source of water in coastal areas. Using RO for the desalination of
brackish groundwater is a cost-effective method and is competitive with seawater
desalination in terms of technical, economical and environmental aspects. RO produces
high quality water which is friendly to the public health and the environment, the soil and
the groundwater. The cost of desalinated brackish groundwater is dropping because of
improvement in membrane technologies. Therefore, a small-scale RO brackish
groundwater desalination plant is selected as a desalination method in the proposed ADR
methodology to treat the abstracted brackish water and recharge it again to the aquifer to
prevent the intrusion of saltwater. A combination of abstraction and recharge wells in
addition to small-scale RO brackish groundwater desalination plant is a cost efficient
system to control saltwater intrusion in coastal areas as it produces freshwater from
brackish water to recharge the aquifer in order to prevent saltwater intrusion and the
produced pure water can be used in domestic, irrigation and industry. A simulation-
optimization model has been developed and will be discussed in the following chapters to
determine the optimum cost of the ADR system considering the optimal well locations,
depths and abstraction/recharge rates.
Chapter (3) Governing equations of coupled fluid flow and solute transport
50
Governing equations of coupled fluid flow and solute transport
3.1 Introduction
Groundwater is an important natural resource. Many agricultural, domestic and industrial
water users rely on groundwater as the sole source because of its low cost and high quality.
However, in recent years it has become clear that human activities and climate change have
a negative impact on both quantity and quality of groundwater resources. The depletion of
groundwater may occur due to excessive pumping and contamination of the groundwater
by waste disposal or other activities. Coupled groundwater flow and solute transport
models can be used to assess the impact of existing or proposed activities on groundwater
quantity and quality (Istok, 1989).
This chapter presents background and theoretical formulation of coupled fluid flow and
solute transport in saturated and unsaturated soil. The theoretical equations include the
coupling between water flow, air flow, heat transfer and solute transport in saturated and
unsaturated soil. The flow of moisture is governed by the liquid transfer and vapour transfer
in soil. The flow of air in soil is considered to be a mixture of dry air transfer and water
vapour transfer. The governing equations for both moisture flow and air transfer are based
on the consideration of conservation of mass. The heat transfer in soil is driven by
conduction, liquid convection and latent heat of vaporisation. The governing equation for
heat transfer is based on the consideration of conservation of energy. The theoretical
formulation of thermal and hydraulic behaviour in unsaturated soil will describe coupled
water, air and heat flow in unsaturated soil. The governing equations are obtained in terms
of the three primary variables; pore water pressure ( wu ), pore air pressure ( au ) and
absolute temperature (T ). Then, the governing equations that describe solute transport will
be given for each phase and the mass transfer processes between phases will be discussed.
The effects of different phenomena that govern the process of solute transport such as
advection, diffusion, dispersion, adsorption, chemical reaction and biological degradation
are also integrated into the model.
Chapter (3) Governing equations of coupled fluid flow and solute transport
51
3.2 History of coupled fluid flow and solute transport
The interest in understanding the mechanism and prediction of solute transport through
soils has increased in the last decades due to growing evidence and public concern that the
quality of the subsurface environment is being adversely affected by industrial, municipal,
agricultural activities and climate change. Salinization of groundwater due to saltwater
intrusion is a special category of contaminants which should be studied and controlled
because it increases the salinity of groundwater and then groundwater may become
unsuitable for use. Accurate predictions need to be made regarding the movement of solute
and energy transport in aquifers. This process can be divided into three sections: the fluid
flow, energy transport and solute transport processes. Once the fluid pressure distribution
through the domain is determined, the flow distribution may be obtained. Solute transport
in the groundwater is done through different mechanisms such as; advection, diffusion,
dispersion, adsorption, chemical reaction, and biological degradation. The transport of
energy in the groundwater and solid matrix of the aquifer includes the changes in
temperature due to the change in conduction, convection and latent heat transfer (AL-
Najjar, 2006). In the following sections a review of fluid flow, energy transport and solute
transport and coupling mechanisms is presented.
3.2.1 Fluid flow in saturated/unsaturated soils
The fluid flow in soils includes groundwater flow and ground air flow. Groundwater
hydrology is very important in evaluating the existing situation and maintaining the
development of water resources. The study of groundwater flow has started by Darcy's
experimente (Darcy, 1856), which introduced what is known as Darcy’s law. Among the
early applications of Darcy's law to groundwater, the contributions of Dupuit, Theim,
Forchheimer, and Boussinesq were of particular note (Zheng and Bennett, 2002). A number
of scientists have contributed to the theory of groundwater flow. Theis (1935) and Jacob
(1950) in particular, are widely recognized for providing the basis for much of modern
groundwater hydrology.
Chapter (3) Governing equations of coupled fluid flow and solute transport
52
Over the years, many models have been developed to simulate groundwater flow. Gottardi
and Venutlelli (2001) developed a finite element model to simulate water flow in saturated-
unsaturated soils. The flow domain can be either in a vertical or in a horizontal plane, or in
a 3-D system. Lapen et al. (2005) developed a 2-D steady-state finite element groundwater
flow model to simulate groundwater flow and examined the sensitivity of flow system to
changes in water recharge and aquifer properties. There have been many studies of
movement of water in saturated and unsaturated soils. However, few have focused on the
movement of the air phase. Most of these studies have ignored the influence of the water
movement on the air phase (Binning, 1994). Green et al. (1970) were among the first to
recognize the importance of the air phase in the unsaturated zone and include it in a one
dimensional finite difference model of two-phase flow. However, the focus of many of the
multiphase flow models of water and air movement in the unsaturated zone has been on the
water phase. Recently, there has been a renewed focus on air movement in the unsaturated
zone. To date there have only been a limited number of models that consider only the air
phase. For example, Johnson et al. (1990) assumed constant moisture content to derive
analytical solutions of air movement. Others who followed this approach include Mendoza
and Frind (1990), Rathfelder et al. (1991), Benson et al. (1993) and McCarthy and Johnson
(1993).
Groundwater flow can be studied in two zones separated by the water table. The region of
low saturation values between the ground surface and the water table is called the
unsaturated zone and the region that is found below the water table is called saturated zone
(Gunduz, 2004). Groundwater flow could either be characterized by a three dimensional or
a vertically averaged two dimensional model. A number of modelers have used a three
dimensional representation of the groundwater flow (e.g., Frind and Verge, 1978; Huyakom
et al. 1986 and McDonald and Harbaugh, 1988). Others such as Aral (1990) have preferred
a vertically averaged two dimensional representation. The numerical simulation of the
saturated groundwater flow equation is performed by finite element or finite difference
method. The finite element method has found wide applications in the field of groundwater
modelling and numerous models used the finite element discretisation (Huyakom et al.,
1986 and Aral, 1990). McDonalds and Harbaugh (1988) applied the finite difference
method.
Chapter (3) Governing equations of coupled fluid flow and solute transport
53
The movement of moisture in variably saturated soil is often modelled by Richards'
equation and closed by constitutive relations to describe the relationship among fluid
pressures, saturations and hydraulic conductivities. The unsaturated zone is characterized
by spatially and temporally varying levels of water content below saturation and negative
capillary pressure heads. Over the years, three different forms of Richards' equation have
been widely applied by scientists; pressure head-based equation, moisture content-based
equation and mixed form of the equation with both the pressure head and the water content.
Pressure-head based and moisture content-based equations were used by a number of
researchers (e.g., Huang et al., 1996; Hills et al., 1989; Gottardi and Venutelli, 1993; Pan
and Wierenga, 1997). However, these forms of Richards' equations have limitations in
application to simulate the saturated conditions (AL-Najjar, 2006).
The mixed-form has been proposed to overcome the difficulties associated with both the
pressure-based and the moisture content-based forms of Richards' equation. This method
uses both the moisture content and the pressure head as the dependent variables. Celia et
al. (1990) solved Richard’s equation using a method that employs the mixed form of the
equations to guarantee mass conservation. The method has been shown to be robust and
accurate, but requires a fine spatial and temporal discretization and so is fairly
computationally demanding (Binning, 1994). The mixed form has both the superior mass
conservation characteristics of the moisture content-based equation as well as the unlimited
applicability to both saturated and unsaturated regions of flow that the pressure-based
equation offers (AL-Najjar, 2006). The numerical solution of the mixed form has found
wide applicability in the last decade and many researchers used this form to model the flow
in variably saturated-unsaturated media (e.g., Tocci et al., 1997; Miller et al., 1998;
Williams and Miller, 1999; Zhang and Ewen, 2000 and Zhang et al, 2002). A number of
analytical solutions have been developed for transient infiltration under various boundary
conditions (e.g., Philip and Knight (1991), Wilson and Gelhar (1981) and Van Genuchten
(1980)). Unsaturated drainage of a uniformly wet soil was solved by Sisson et al. (1980)
for gravitational flow, whereas more complicated solutions for drainage with capillary
suction was derived by Warrick et al., (1990) and Philip (1992).
Chapter (3) Governing equations of coupled fluid flow and solute transport
54
3.2.2 Energy transport Energy-transport simulation may be employed to model thermal regimes in aquifers,
subsurface heat conduction, aquifer thermal-energy storage systems, geothermal reservoirs,
thermal pollution of aquifers, and natural hydrogeologic convection systems (Voss and
Provost, 2003). Energy is transported in the water-solid matrix by flow of groundwater and
by the thermal conduction from higher to lower temperature through both the fluid and
solid. In fact, the average velocity of fluid is affected by variation in temperature because
change in temperature leads to the change in density and viscosity and subsequently the
fluid velocity. The simulation of energy transport is based on the change of the amount of
energy stored in the solid matrix and fluid. The stored energy in a volume may change due
to ambient water with different temperature flowing in, well water of a different
temperature injected, changes in the total mass of water in the block, thermal conduction
(energy diffusion) into or out, and energy production or loss due to nuclear, chemical or
biological reactions. The transport of energy in the groundwater and solid matrix of an
aquifer includes the changes in temperature due to the change in conduction, convection
and latent heat transfer (Voss and Provost, 2003).
Energy transport is simulated through numerical solution of an energy-balance equation.
The solid grains of the aquifer matrix and fluid are locally assumed to have equal
temperature, and fluid density and viscosity may be affected by the temperature. A number
of numerical models have been developed to simulate fluid flow and energy transport in
soil. Dakshanamurthy and Fredlund (1981) proposed a one dimensional formulation for
coupled heat, mass and air flow in unsaturated soils. Three partial differential equations
were employed to describe the three flow phases in an unsaturated soil. The differential
equation for air flow was based on Fick's law and the differential equation for fluid and
water vapour flow (moisture flow) was based on Darcy's law. Water vapour flow due to
diffusion and advection processes was also considered. A Fourier diffusion equation was
employed to describe heat flow in unsaturated soils. Heat flow due to conduction and latent
heat transfer caused by phase changes were also considered in the heat flow. A finite
difference scheme was applied to solve the governing equations and the proposed model
was applied to simulate the coupled flow behavior of compacted Regina clay.
Chapter (3) Governing equations of coupled fluid flow and solute transport
55
3.2.3 Solute transport in saturated/unsaturated soils
In recent years, great efforts have been directed towards establishing better knowledge of
what governs solute transport in soils. Solute transport was considered only in a very
limited way in early groundwater investigations. The method of analysis generally applied
was advective calculation, in which solutes were assumed to move at the calculated average
velocity of the groundwater, and to be unaffected by sorption, kinetic reaction, and other
processes. The transport problem in porous media was the subject of extensive research in
the chemical engineering field. Within the field of groundwater, it was recognized that the
average fluid velocity did not describe the actual motion of individual solute particles, and
that advective calculation could therefore never give a full description of solute movement
(AL-Najjar, 2006). A number of laboratory and theoretical investigations of dispersion
were done by Day (1956) and Rifai et al. (1956). The theory of advective-dispersive
transport continued to develop by Bear (1961) and Bachmat (1967). A number of analytical
solutions to the advective-dispersive transport equation were developed in this period;
several of these are summarized by Ogata (1970).
The modelling of transport in groundwater flow has been done using different analytical
and numerical methods. Analytical solutions to selected groundwater flow and solute
transport problems are discussed in Bear (1979). Finite element codes have also been used
effectively to simulate transport of subsurface contaminants (Bear et al., 1993; Istok, 1989).
Numerous mathematical models were developed for simulating groundwater flow and
solute transport. Nwaogazie (1986) developed a two dimensional transient finite element
code for modelling groundwater flow and transport. Ahuja and Lehman (1983) and
Wallach et al. (1989) applied a diffusive transport model to predict the solute transport in
soil in the absence of infiltration. Snyder and Woolhiser (1985) conjectured that using a
convective-dispersive model of solute transport in soils would be most appropriate when
infiltration occur. Their model was used by Wallach and Shabtai (1992) to describe the
downward solute movement in soil in the presence of surface water. Zienkiewicz and
Codina (1995) applied the characteristic Galerkin method to the advection-diffusion
problems and developed an explicit algorithm for these problems.
Chapter (3) Governing equations of coupled fluid flow and solute transport
56
Smith et al. (1992) developed a non-equilibrium sorption dispersion-advection model that
involves a convolution integral of the product of the rate of change of concentration and a
time dependent sorption coefficient. Sheng and Smith (2002) presented a two dimensional
finite element method for the solution of advection-dispersion transport equation for multi
component contaminants. Their analysis was restricted to nonlinear, equilibrium controlled
sorption and exchange of soluble inorganic ions. Zhang and Brusseau (2004) presented a
distributed domain mathematical model incorporating the primary mass transfer processes
that mediate the transport of immiscible organic liquid constituents in water-saturated,
locally heterogeneous porous media. Specifically, the impact of grain heterogeneity on
immiscible liquid dissolution and sorption and desorption was represented in the model by
describing the system as comprising a continuous distribution of mass-transfer domains.
Kacur et al. (2004) discussed the numerical approximation schemes for the solution of
contaminant transport with adsorption in dual-well flow. Their method was based on time
stepping and operator splitting for the transport with adsorption and diffusion. The
nonlinear diffusion was solved by a finite volume method and by Newton's type of
linearization.
Remesikova (2005) introduced an efficient operator splitting scheme for solving two
dimensional convection-diffusion problems with adsorption. He particularly, considered a
practical problem of soil parameters identification using dual-well tests by using a general
mathematical model including contaminant transport, mechanical dispersion and molecular
diffusion and adsorption in both equilibrium and non-equilibrium models. However, due to
the transformation, the transport problem (linear or non-linear) was reduced to one
dimensional and solved in an analytical form. The dispersion part was solved using
standard finite volume method. In spite of all the advances achieved and new techniques
developed, there is no single technique that can yield completely satisfactory solutions to
the numerical solution of advection-dominated contaminant transport. It remains a difficult
problem due to the often contradictory needs to suppress numerical dispersion, avoid
artificial oscillation and conserve mass (Gunduz, 2004).
Chapter (3) Governing equations of coupled fluid flow and solute transport
57
In the recent years, there has been a profusion of studies of the transport equations in the
unsaturated zone. The impetus for this rapid development has been the strict regulatory
stance on groundwater pollution (Gee et al., 1991). Therefore, research on solute transport
in the unsaturated zone has become more important and the studies for developing
numerical models have increased. A number of analytical methods were developed to
simulate water movement and solute transport in unsaturated zones (Islas and Illangasekare,
1992 and Barry et al., 1993). Smiles et al. (1978) developed a quasi-analytical solution for
non-reactive solute flow during unsteady horizontal infiltration under constant
concentration boundary conditions which has been discussed by Watson and Jones (1981)
particularly in relation to assessing the performance of the solute model. Lessoff and
Indelman (2004) presented an analytical model of advective solute transport by unsteady
unsaturated gravitational infiltration.
Finite difference techniques are the simplest of the numerical techniques to apply and there
are a large number of models using this solution technique. Weeks et al. (1982) were some
of the first to use a numerical model to study transport in the unsaturated zone. The finite
difference models range form the simple one dimensional model of Rosenbloom at al.
(1993) through to the comprehensive model of Sleep and Sykes (1993). The finite
difference models have usually used forward (explicit) time stepping. Benson et al. (1993)
and Sleep and Sykes (1993) presented a summary of the arguments for the different forms
of temporal discretization. The implicit method is more computationally expensive than the
explicit method. The implicit time stepping scheme is unconditionally stable, so that large
time steps can be taken. However, large time steps may not be advantageous in an implicit
scheme as the truncation error increases with time step size leading to a loss of accuracy in
the solution. In contrast, the explicit methods do have a limitation on the time step size, so
that small time steps must be taken. Accuracy of an explicit scheme increases with
increasing time step size.
Another common numerical approach to solving the transport equations has been to use the
finite element method. The Galerkin finite element method was applied by Mendoza and
Frind (1990) and Culver et al. (1991). The solutions obtained by the method are prone to
oscillations. Karkuri and Molenkamp (1997) analyzed the advection-dispersion of non-
reactive pollutant movement through a layered porous medium domain under the effect of
Chapter (3) Governing equations of coupled fluid flow and solute transport
58
transient groundwater flow. The governing partial differential equations of the groundwater
flow and advection-dispersion of pollutant together with their integral formulations were
based on Galerkin's method and Green's theorem. Li et al. (1999) presented a numerical
model to simulate miscible contaminant transport through unsaturated soils to account for
the influence of multiple non-equilibrium sources on the contaminant transport. Marshall et
al. (2000) developed a general methodology for investigating the effect of short-term
temporal variability in infiltration on the long term transport of contaminants in soils. A one
dimensional, unsaturated transport model was used to simulate the transport of a sorbing,
non-volatile solute, using either steady state or randomly varying infiltration. Islam and
Singhal (2002) presented a one dimensional reactive multi-component landfill leachate
transport model coupled to three modules (geochemical equilibrium, kinetic
biodegradation, and kinetic precipitation–dissolution) to simulate the migration of
contaminants in soils under landfills. It was found that, the transport of contaminants
through unsaturated soil is influenced by several processes, two of the most important
being sorption and transformation.
Worch (2004) presented a solute transport model that describes non-equilibrium adsorption
in soil and groundwater systems by mass transfer equations for film and intraparticle
diffusion. Furthermore, a sensitivity analysis was performed to illustrate the influence of
different process and sorption parameters on the shape of the calculated breakthrough
curves. Kacur et al. (2004) described a numerical approximation scheme for the solution of
contaminant transport problems with diffusion and adsorption in equilibrium and non-
equilibrium modes. Kacur et al. (2004) method was based on time stepping and operator
splitting. The non-linear transport was solved semi-analytically via the multiple Riemann
problems, the non-linear diffusion by a finite volume method and by Newton’s type of
linearization. Finally the reaction part, incorporating the non-equilibrium adsorption, was
transformed to an integral equation which was solved numerically using time discretization.
Several researchers have tried to improve the finite element solution by adding upstream
weighting. Upstream weighting was demonstrated to be effective in reducing oscillations
in the saturated transport equation by Dick (1983).
Chapter (3) Governing equations of coupled fluid flow and solute transport
59
3.2.4 Coupling mechanisms
Over the years, a number of models have been developed for coupling fluid flow, heat
transfer and solute transport. Couvillion and Hartley (1985) investigated the movement of
thermally induced drying fronts in sandy soils and presented an explicit finite difference
solution of three governing equations describing the flow of heat, moisture and air.
Geraminegad and Saxena (1986) presented a mathematical model for the flow of heat,
moisture and gas in unsaturated porous media. Three linear partial differential equations
were derived; one for heat flow based on a modified Philip and de Vries approach, one for
liquid flow based on a modified Philip and de Vries approach and one for gas flow based
on Darcy's law.
Pollock (1986) developed a mathematical model to describe the coupled transport of
energy, liquid water and water vapour, and dry air transport in unsaturated media. Three
coupled non-linear partial differential equations were developed, each describing one flow
phase. Connell and Bell (1993) developed a numerical model to describe coupled liquid and
vapour flow processes in waste dumps under climatic influences. Three governing
equations were developed to describe the flow of liquid, air and heat. Thermodynamic
equilibrium between liquid and vapour phases was not assumed for this model. Instead,
separate relationships were employed for vapour (viscous vapour and diffusive vapour) and
liquid phases. Pinder and Abriola (1986) solved the fully coupled three phase problem
using an implicit finite difference model with a Newton Raphson non-linear iteration
scheme with the assumption of a constant-pressure immobile gas. Zhou and Rowe (2005)
analysed a set of fully coupled governing equations expressed in terms of displacement,
capillary pressure, air pressure, and temperature increase and solved the equations using the
finite element method with a mass conservative numerical scheme. Gawin et al. (2005)
described a fully coupled numerical model to simulate the flow transient phenomena
involving heat and mass transfer in deforming porous media. The governing differential
equations were solved simultaneously in terms of displacements, temperature and capillary
pressure using the finite element method.
Chapter (3) Governing equations of coupled fluid flow and solute transport
60
Thomas and Ferguson (1999) presented a coupled heat and mass transfer numerical model
describing the migration of a contaminant gas through unsaturated porous medium. The
model treats the migration of liquid water, air, heat and contaminant gas separately with
independent system variables of capillary potential, temperature, pore air pressure and
concentration of the contaminant gas. Sheng and Smith (2000) presented a characteristic
finite element algorithm for modelling contaminant transport problems coupled with
nonlinear adsorption. Crowe et al. (2004) developed a numerical model for simulating
groundwater-wetland interactions and contaminant transport. The model calculated
transient hydraulic head in a two dimensional heterogeneous domain.
Liang et al. (2002) studied the transport mechanism of hydrophobic organic chemicals and
the energy change in a soil/solvent system. A soil leaching column chromatographic
experiment at an environmental temperature range of 20-40o C was carried out. It was
found that the transport process quickens with the increase of column temperature. Gao
(2001) presented a model for simulating the transport of chemically reactive components in
conjunction with energy transport in saturated and unsaturated groundwater systems.
McGrail (2001) developed a numerically based simulator to assist in the interpretation of
complex laboratory experiments examining transport processes of chemical and biological
contaminants subject to nonlinear adsorption or source terms. Javadi and Al-Najjar (2008)
and Javadi et al. (2008) presented a numerical model for the simulation of water and air and
contaminant transport through unsaturated soils considering chemical reactions and
assuming constant density flow.
Saltwater intrusion in coastal aquifers has been simulated using different methods. Coupled
fluid flow and solute transport models can help in understanding the mechanism of
saltwater intrusion and can be applied in the prediction of solute concentration in coastal
aquifers. A review of fluid flow, solute transport, and coupling mechanisms has been
presented. The following sections present the development of coupled fluid flow and solute
transport model to simulate saltwater intrusion in coastal aquifers.
Chapter (3) Governing equations of coupled fluid flow and solute transport
61
3.3 The governing equation for the moisture transfer
The moisture transfer in unsaturated soils is composed of two main phase movements:
water transfer and water vapour transfer. The volumetric moisture content in unsaturated
soil, , is defined as the sum of volumetric water content, w , and volumetric vapour
content, v :
vw (3.1)
The law of conservation of mass for water phase states that the temporal discretisation of
the water content must be equal to the spatial discretisation of the water flux and this can be
written as (Ramesh, 1996):
Fvt wwww
w (3.2)
where,
w : is the density of water [M][L]-3,
t : is the time [T],
wv : is the velocity of water [L][T]-1,
F : is the source or sink term.
Similarly, the law of conservation of mass for vapour phase states that the temporal
discretisation of the vapour content must be equal to the spatial discretisation of vapour flux
and this can be written as (Ramesh, 1996):
Fvvt wavvwv
w )( (3.3)
where,
v : is the density of water vapour [M][L]-3.
Chapter (3) Governing equations of coupled fluid flow and solute transport
62
av : is the velocity of the air [L][T]-1.
vv : is the velocity of the water vapour [L][T]-1.
By substituting equations (3.2) and (3.3) into (3.1), the conservation of mass for moisture
transfer can be written as:
0)()()()()(vawvww
vaww vvvtt
(3.4)
The volumetric water content w and the volumetric air content a can be expressed in
terms of porosity n and degree of saturation as following:
ww nS (3.5)
aa nS (3.6)
wa SS 1 (3.7)
where,
wS : is the degree of water saturation in unsaturated soil and
aS : is the degree of air saturation in unsaturated soil.
Substituting equations (3.5) and (3.6) into (3.4) the governing equation for the moisture
transfer can be expressed as:
0)()()()()(vawvww
vaww vvvt
nSt
nS (3.8)
From equation (3.8) the flow of water is governed by the pressure head gradient and a
gradient due to elevations differences and vapour transport is governed by the diffusive and
pressure flows. Therefore, the various components of each flow phase and the flow laws of
the two phases of moisture transfer, namely, water flow and vapour flow are explained
separately in next sections.
Chapter (3) Governing equations of coupled fluid flow and solute transport
63
3.3.1 Water flow
The hydraulic head gradient is defined as the sum of the pressure head and the elevation
head gradients, which is considered the potential driving force for water flow (Fredlund and
Rahardjo, 1993). According to Darcy’s law the velocity of water wv can be expressed as
(Darcy, 1856):
zuKvw
www (3.9)
where,
wK : is the hydraulic conductivity of water [L][T]-1.
wu : is the pore water pressure (hydraulic pressure) [M][L]-1[T]-2 .
w : is the unit weight of water [M][L]-2[T]-2 .
z : is the unit normal oriented downwards in the direction of the force of gravity.
The relation between pore water pressure and hydraulic head can be expressed as (Fredlund
and Rahardjo, 1993):
zg
uhw
ww (3.10)
where,
wh : is the hydraulic head [L]
g : is the gravitational acceleration [L][T]-2.
z : is the elevation above the datum (positive upwards) [L].
The hydraulic conductivity can be expressed as (Fredlund and Rahardjo, 1993):
w
www
gkK (3.11)
Chapter (3) Governing equations of coupled fluid flow and solute transport
64
where,
wk : is the effective permeability of water [L]2
w : is the dynamic viscosity of water [M][L]-1[T]-1
Kaye and Laby (1973) expressed the dependence of dynamic viscosity of liquid water w
(Ns/m2) on absolute temperature T in the form of a relation as:
003562.1 5.010*)229(2.661)( TTw (3.12)
The hydraulic conductivity of unsaturated soil is influenced by a number of factors such as
mineralogical composition, particle sizes and the size distribution, void ratio e and the
fluid characteristics (Mitchell, 1976). Hydraulic conductivity is a function of any two of
three possible volume-mass properties (Lloret and Alonso, 1980 and Fredlund, 1981) as
following:
),( www SeKK (3.13)
),( www eKK (3.14)
),( wwww SKK (3.15)
In general, the hydraulic conductivity is significantly affected by the combined change in
void ratio and degree of saturation of the soil (Fredlund and Rahardjo, 1993). Matyas and
Radhakrishna (1968) proposed that the water degree of saturation wS could be expressed as
stress dependent quantity and initial void ratio, initial degree of saturation and stress
parameters, namely, net stress, deviatoric stress, and suction. Alonso et al. (1988) and
Fredlund and Rahardjo (1993) pointed out that the effect of changes in stress on the degree
of saturation is insignificant and can be negligible. Therefore, the major effect on the
degree of saturation comes from the changes in suction. Suction, s, is defined as the
difference between pore air pressure au and pore water pressure wu and is mathematically
represented as:
Chapter (3) Governing equations of coupled fluid flow and solute transport
65
)( wa uus (3.16)
The soil suction was referred to as the free energy state of soil water (Edlefsen and
Anderson (1943) and Fredlund and Rahardjo (1993)). Thus, for unsaturated soils, suction
is dependent on the surface energy , which is in turn, dependent on absolute temperature
T . Therefore suction s at any moisture content w and temperature T can be obtained by
the following relationship:
),()(
)(),( rwrrr
w TsTTTs (3.17)
where,
rT : is the reference temperature.
rs : is the suction at the reference temperature.
r : is the surface energy at the reference temperature.
: is the surface energy at the given temperature T .
Edlefsen and Anderson (1943) proposed a relation between the state surface energy and
the absolute temperature T as:
T00001516.01171.0 (3.18)
Incorporating the above mentioned variation of suction with temperature, the degree of
saturation of water can be expressed as:
),( TsSS ww (3.19)
or in differential form as:
tT
TS
ts
sS
tS www (3.20)
Chapter (3) Governing equations of coupled fluid flow and solute transport
66
By substituting equation (3.16) into equation (3.20)
tT
TS
tu
sS
tu
sS
tS wwwaww (3.21)
3.3.2 Water vapour flow
The vapour transfer in unsaturated soils occurs as a result of diffusive and pressure flows.
The air phase is in bulk motion due to free or forced convection (Rohsenow and Choi
1961). This bulk air is considered to be a binary mixture of water vapour and dry air. Soil
water content has a significant effect on the air permeability. In general, higher water
contents reduce the air-filled porosity, thereby decreasing the connected pores through
which air can flow by advection. The part of vapour transferred by the pressure flow is
governed by the movement of the air mixture and can be described using the extension of
Darcy’s law. The part of vapour transferred by diffusion can be interpreted using the theory
of Philip and de Vries (1957). Fick’s (1855) and Darcy’s laws have been commonly used to
describe the flow of air through a porous media. According to generalised Darcy’s law in
unsaturated soil, the velocity of pore air av can be defined as:
aaaa
aa uKukv (3.22)
where,
ak : is the effective permeability of air [L]2
a : is the absolute viscosity of air [M][L]-1[T]-1
aK : is the unsaturated conductivity of air [L][T]-1 , which can be defined as following:
a
aaa
gkK (3.23)
where, a : is the density of air [M][L]-3
Chapter (3) Governing equations of coupled fluid flow and solute transport
67
The air properties can be considered to be constant and the only properties that affect the
conductivity of air are the volume-mass properties such as the degree of saturation and the
void ratio (Fredlund and Rahardjo, 1993):
),( aaa SeKK (3.24)
Philip and de Vries (1957) proposed an expression for the vapour velocity vv as:
vw
avvatmsv
VDv (3.25)
where,
atmsD : is the molecular diffusivity of vapour through air.
vV : is the mass flow factor.
v : is the vapour torousity factor.
v : is the vapour density gradient.
The determination of these parameters which describe the vapour velocity is discussed
below.
Philip and de Vries (1957) adopted the approach suggested by Krischer and Rohnalter
(1940) to determine the molecular diffusivity atmsD (Ramesh, 1996), where the diffusion
due to the temperature gradient was represented by the expression:
a
atms uTD
23610*893.5 (m2/sec) (3.26)
where, T is the absolute temperature. This equation is valid in the temperature range of
(20 – 70 oC , 293 - 343 oK ).
Chapter (3) Governing equations of coupled fluid flow and solute transport
68
The mass flow factor vV is evaluated as suggested by Philip and de Vries (1957). They
adopted the expression suggested by Partington (1949), who showed that for steady state
diffusion in closed system between evaporating source and a condensing sink, the mass
flow factor vV can be expressed as:
a
vav u
uuV (3.27)
where vu is the pore water vapour pressure [M][L]-1[L]-2, which can be calculate as:
TRu vvv (3.28)
where,
vR : is the specific gas constant for water vapour [L]2[T]-2[K]-1 (461.5J/kg K).
Philip and de Vries, (1957) suggested that expression for mass flow factor in equation
(3.27) may not be valid under non-stationary conditions, but the value is quite close to 1
under normal soil temperatures.
Edlefesen and Anderson (1943) proposed a thermodynamic relationship for the density of
water vapour v as:
TRghv
v exp00 (3.29)
where h is the relative humidity, is the capillary potential, which is given by:
w
aw uu (3.30)
Chapter (3) Governing equations of coupled fluid flow and solute transport
69
and 0 is the saturated soil vapour density [M][L]-3 and can be calculated by a relationship
was given by Ewen and Thomas (1989) as:
123
0})273(10*1634.0)273(06374.0exp{4.1941 TT (3.31)
The density of pore vapour v is dependent on the saturated soil vapour density 0 (which
is dependent on the absolute temperature T ) and the relative humidity h (which is
dependent on both suction s and temperature T ). Thus, the gradient of vapour density
v can be represented as:
hhv 00 (3.32)
By expanding the terms
TThs
shT
Thv 0
0 (3.33)
By substituting suction from equation (3.16) into equation (3.33) and rearrange the terms,
the gradient of vapour density v can be written as:
TTh
Thuu
sh
wav 00
)0 ( (3.34)
or,
TKuKuK fTafawfwv (3.35)
where,
shK fw 0 (3.36)
shK fa 0 (3.37)
Chapter (3) Governing equations of coupled fluid flow and solute transport
70
Th
ThK fT 0
0 (3.38)
Substitute equation (3.35) into equation (3.25) the velocity of water vapour can be written
as:
TKVDuKVDuKVDv fTw
avvatmswfw
w
avvatmsafa
w
avvatmsv (3.39)
Philip and de Vries (1957) indicated that the theory proposed in the equation (3.39) could
not satisfactorily predict vapour transfer under a thermal gradient. Two refinements were
made to the thermal gradient term (Ramesh, 1996):
Adding a flow area factor f in an attempt to reduce the vapour flow with
increasing reduction in the available flow area.
Introducing a pore temperature gradient T
T a)( which is the ratio of the average
temperature gradient in the air filled pores to the overall temperature gradient.
Introducing these two refinements, the expression for vapour velocity in equation (3.39)
can be written as:
TKT
TVDfuKVDuKVDv fTw
avatmswfw
w
avvatmsafa
w
avvatmsv
)( (3.40)
Preece (1975) suggested an expression of pore temperature gradient (Ramesh, 1996) which
has been implemented in the present work and is defined as:
)21(11
12
31)(
v
aGBBGT
T (3.41)
Chapter (3) Governing equations of coupled fluid flow and solute transport
71
where,
1)(
w
vaB (3.42)
09.00)09.0(325.033.011.110033.0
09.0)(325.03333.0
ww
ww
V
nn
nn
n
G (3.43)
where,
w : is the thermal conductivity of water [M][L][T]-3[K]-1 .
a : is the thermal conductivity of air [M][L][T]-3[K]-1 .
v : is the thermal conductivity of vapour [M][L][T]-3[K]-1 which can be expressed as:
thLVD vatmsv
0 (3.44)
Ewen and Thomas, (1989) suggested two alternatives to extend vapour velocity equation
proposed by Philip and de Vries, (1957) these alternatives are:
The vapour flow area factor f should be present in both temperature and moisture
gradient terms.
The flow area factor ( av ) can be modified to be equal to the porosity to avoid
choking occurrence of vapour flow at high moisture contents.
By including these to changes and rewriting the equation (3.40), the vapour velocity can be
written as:
TKT
TnVDuKnVDuKnVDv fTw
avatmswfw
w
vatmsafa
w
vatmsv
)( (3.45)
Chapter (3) Governing equations of coupled fluid flow and solute transport
72
or,
TKuKuKv vTavawvwv (3.46)
where,
fww
vatmsvw KnVDK (3.47)
faw
vatmsva KnVDK (3.48)
fTw
avatmsvT K
TTnVDK )( (3.49)
3.3.3 Mass balance equation for the moisture transfer
The various components of liquid transfer and vapour transfer have been discussed in the
previous sections. These components are used to derive the governing equation for the
moisture transfer. Substituting equations (3.7), (3.9), (3.22) and (3.46) into equation (3.8)
the mass conservation equation for the moisture flow can be written as:
0)()(
)()}1{()(
vaawvTwvwava
ww
ww
vwww
uKTKuKuK
zuKtSn
tnS
(3.50)
Equation (3.50) comprises various terms, Therefore, simplifying each term is necessary in
order to make the governing equation of moisture flow written in terms of the three primary
variables pore water pressure wu , pore air pressure au and absolute temperature T . The
simplification is explained below:
1. Applying product rule of differentiation to the first and second terms of equation (3.50)
will yield:
Chapter (3) Governing equations of coupled fluid flow and solute transport
73
tSn
tnS vwww )}1{()(
)1()1(
)1( wv
vw
vwww
ww Snt
ntS
Stnn
tS
Stn
tSn w
vw )( +t
nS va + )( vaww SS
tn
tSn w
vw )( +t
nS va
2. The third term of equation (3.50) can be rewritten as:
)( ww
ww zuK )()( zKuK wwww
w
w
3. The fourth term of equation (3.50) can be rewritten as:
)( wvTwvwava TKuKuK TKuKuK vTwvwavaw
4. The fifth term of equation (3.50) can be rewritten as:
)( vaa uK )( aav uK
By substituting the simplified terms into equation (3.50), it can be written as:
tSn w
vw )( +t
nS va )( ww
w
w uK )( aav uK
TKuKuK vTwvwavaw = )( zKww (3.51)
Substituting equation (3.21) into the first term of equation (3.51) yields:
tT
TS
tu
sS
tu
sSn
tSn wwwaw
vww
vw )()(
tT
TS
tu
sS
tu
sSn wawww
vw )(
Chapter (3) Governing equations of coupled fluid flow and solute transport
74
tuC
tTC
tuC
tSn a
fafTw
fww
vw )( (3.52)
where,
sSnC w
vwfw )( (3.53)
TSnC w
vwfT )( (3.54)
sSnC w
vwfa )( (3.55)
Substituting equation (3.35) into the second term of equation (3.51) yields
tnS v
a tuC
tTC
tuC a
vavTw
vw (3.56)
where,
fwavw KnSC (3.57)
fTavT KnSC (3.58)
faava KnSC (3.59)
Substituting equation (3.52) and (3.56) into equation (3.51) yields the governing equation
of moisture flow written in terms of the three primary variables wu , au and T :
)( www
wavavT
wvw
afafT
wfw uK
tuC
tTC
tuC
tuC
tTC
tuC
)( aav uK TKuKuK vTwvwavaw = )( zK ww
or
tuC
tTC
tuC a
wawTw
ww = wawawTwww JuKTKuK ][][][ (3.60)
Chapter (3) Governing equations of coupled fluid flow and solute transport
75
where,
vwfwww CCC (3.61)
vTfTwT CCC (3.62)
vafawa CCC (3.63)
wvww
wwww KKK (3.64)
vawavwa KKK (3.65)
vTwwT KK (3.66)
)( zKJ www (3.67)
3.4 The governing equation of air transfer
The transfer of air through the soil is governed by two main effects (Alonso et al., 1988):
the bulk air flow due the pressure gradient, which can be determined by the use of Darcy’s
law and the dissolved air transfer within pore-liquid, which can be determined by the use of
Henry’s law. The governing differential equation of air transfer is based on the
conservation of mass principle, which dictates that the temporal discretisation of the dry air
is equal to the spatial discretisation of the dry air flux. This can be expressed as:
0dawaadawaa vHv
tH (3.68)
where,
aH : is the Henry’s volumetric coefficient of solubility.
da : is the density of dry air [M][L]-3.
Substituting equations (3.5) and (3.6) for w and a , and rewriting equation (3.68) in
terms of degree of saturation and porosity, the equation for air flow can be written as:
Chapter (3) Governing equations of coupled fluid flow and solute transport
76
0dawaadawaa vHv
tnSHnS (3.69)
The various parameters involved in equation (3.69) for describing the air flow are discussed
below.
Pollock (1986) adopted an approach to determine the air density. He considered the air
transfer in unsaturated soils to be a mixture of dry air transfer and water vapour transfer.
This mixture obeys all the laws of ideal gases which can be applied for both dry air and
water vapour (Geraminagad and Saxena, 1986). The application of the ideal gas law to dry
air leads to:
TRu dadada (3.70)
where,
dau : is the pore dry air pressure [M][L]-1[T]-2.
daR : is the specific gas constant of dry air [L]2[T]-2[K]-1.
Applying Dalton’s law of partial pressures to the pore-air mixture yields:
vdaa uuu (3.71)
To determine the dry air density, substituting equations (3.28) for vu and (3.70) for dau into
(3.71) and rearranging yields:
)()( TRTRu vvdadaa
vda
v
da
ada R
RTR
u (3.72)
The time derivative of dry air density can then be written as:
Chapter (3) Governing equations of coupled fluid flow and solute transport
77
tRR
tT
TRu
tu
TRtv
da
v
da
aa
da
da2
1 (3.73)
By substituting equations (3.9) for the velocity of water wv and (3.22) for the velocity of
air av into (3.69) and rearranging the terms, the equation for dry air can be written as:
0)( daw
wwaaa
dawaa zuKHuKt
SHSn (3.74)
Equation (3.74) comprises various terms, Therefore, simplifying each term is necessary in
order to make the governing equation of dry air flow written in terms of the three primary
variables wu , au and T . The terms can be simplified as follows:
1. Applying product rule of differentiation to the first term of equation (3.74) yields:
tnSHS
tSHSn
tSHSn
tnSHnS
waadada
waawaa
dadawaa )()()(
The first term on the right hand side of the above equation can be written as:
tSnH
tSn
tSHSn w
daaa
dawaa
da)(
Substituting equation (3.7) for the degree of saturation of air aS into this equation will
result:
tSnH
tSn
tSHSn w
daaw
dawaa
da)1()(
tSHn w
ada )1(
As a result, the first term of the equation (3.74) can be rewritten as:
Chapter (3) Governing equations of coupled fluid flow and solute transport
78
tSHn w
ada )1( + t
SHSn dawaa )( +
tnSHS waada )(
2. The second term of equation (3.74) can be rewritten as:
daw
wwaaa zuKHuK )(
= daw
wwwaaa
zuKHuK )(
= ))(()( wwww
daaadaa zuK
HuK
By substituting the simplified terms into equation (3.74), it can be written as:
tSHn w
ada )1( +t
SHSn dawaa )( +
tnSHS waada )(
))(()( wwww
daaadaa zuKHuK (3.75)
By substituting equation (3.21) into the first term of (3.75) and rearranging,
tT
TS
tu
sS
tu
sSHn
tSHn wwwaw
adaw
ada )1()1(
The first term of the above equation can be written in a simple form in terms of the three
unknowns as:
tuC
tTC
tuC
tSHn a
aaaTw
aww
ada 111)1( (3.76)
where,
Chapter (3) Governing equations of coupled fluid flow and solute transport
79
sSHnC w
adaaw )1(1 (3.77)
TSHnC w
adaaT )1(1 (3.78)
sSHnC w
adaaa )1(1 (3.79)
By substituting equation (3.35) into equation (3.73), the time derivative of the dry air
density can be written as:
tu
fatT
fttu
fwda
v
da
aa
da
da aw KKKRR
tT
TRu
tu
TRt 21
Rewriting the right hand side terms in a simple form in terms of the three unknowns yields:
tuC
tTC
tuC
ta
daadaTw
dawda (3.80)
where,
fwda
vdaw K
RRC (3.81)
fTda
vadaT K
RR
TC (3.82)
fada
v
dadaa K
RR
TRC 1 (3.83)
By substituting equation (3.80) into the second term of equation (3.75), the second term of
equation (3.75) can then written as:
tu
CtTC
tu
CSHSnt
SHSn adaadaT
wdawwaa
dawaa )()(
Rewriting in a simple form in terms of the three unknowns:
Chapter (3) Governing equations of coupled fluid flow and solute transport
80
tuC
tTC
tuC
tSHSn a
aaaTw
awda
waa 222)( (3.84)
where,
dawwaaaw CSHSnC )(2 (3.85)
daTwaaaT CSHSnC )(2 (3.86)
daawaaaa CSHSnC )(2 (3.87)
Finally, substituting equations (3.76) and (3.84) into equation (3.75) leads to the governing
differential equation for the air transfer in unsaturated soils:
tuC
tTC
tuC
tuC
tTC
tuC a
aaaTw
awa
aaaTw
aw 222111
awawaaa JuKuK )(
tuC
tTC
tuC a
aaaTw
aw = aaaawaw JuKuK )( (3.88)
where,
21 awawaw CCC (3.89)
21 aTaTaT CCC (3.90)
21 aaaaaa CCC (3.91)
ww
daaaw KHK (3.92)
wdaaaa uKK (3.93)
)( zKHJ wdaaa (3.94)
Chapter (3) Governing equations of coupled fluid flow and solute transport
81
3.5 The governing equation of heat transfer
There has been a continuous increase in temperature of earth due to climate change which
is resulting in thermal expansion of oceans and seas. The change in temperature could have
an effect on solute transport as the fluid viscosity and density will change which will affect
on the solute migration in the aquifer. The effect of temperature change on saltwater
intrusion is very important but it has not been studied in the present work and has been
proposed as one of the subjects for future research. The mathematical formulation has been
added here because temperature is included as one of the degrees of freedoms in the current
model. The governing equation of heat transfer is derived based on the effects of three
components: conduction, liquid convection and latent heat of vaporisation. The radiation
effects are assumed to be negligible and hence are not included in the governing equation
for heat transfer. The governing differential equation for heat transfer is derived based on
the principle of conservation of energy. The conservation of energy for heat flow states
that the temporal change in heat content must be equal to the special change in heat flux
Q and can be written mathematically as:
0Qt
(3.95)
The heat content of unsaturated soils is considered to be the sum of two components; soil
heat storage capacity and the contribution resulting from the latent heat of vaporisation.
Therefore, the latent heat energy can be written as:
varc nLSTTH )( (3.96)
where,
T , :rT are the given and the reference temperatures respectively.
L : is the latent heat of vaporisation [L]2[T]-2.
cH : is the heat capacity of the soil at the reference temperature which is defined as:
)()1( daapdavapvwwpwspsc SCSCSCnCnH (3.97)
Chapter (3) Governing equations of coupled fluid flow and solute transport
82
where,
s : is the density of solid particles [M][L]-3.
pdapvpwps CCCC ,,, : are the specific heat capacities of solid particles, liquid, vapour and
dry air respectively [L]2[T]-2[K]-1.
The heat flux per unit area Q can be defined as (Ramesh, 1996):
))(()( rdaapdavapvwvpvwwpwvawvT TTvCvCvCvCLvvTQ (3.98)
Where T is the intrinsic thermal conductivity of soil [M][L][T]-3[K]-1.
The thermal conductivity of unsaturated soil has been found to be a function of degree of
saturation and can be expressed as (Ewen and Thomas, 1987):
)( wTT S (3.99)
The heat flux equation (3.98) includes the effects of three components (Ramesh, 1996):
1- Conduction.
2- Latent heat flow in soil water vapour arising from vapour flow and
3- The rate of heat flow due to convection of sensible heat in (i) the liquid phase, (ii) the
vapour phase associated with flow induced by vapour pressure gradient, (iii) the vapour
phase movement associated with the bulk flow of air and (iv) the air phase.
Substituting equations (3.96) for , (3.97) for cH and (3.98) for Q into (3.95), ignoring
the term corresponding to soil deformation, the governing equation for heat transfer can be
written as:
0)()()()()()()(
)(
radapdaravpvrvwpv
rwwpwavvwT
va
av
crc
TTvCTTvCTTvCTTvCvLvLT
tnLS
tSnL
tHTT
tTH
(3.100)
Chapter (3) Governing equations of coupled fluid flow and solute transport
83
The various terms of the governing equation of heat transfer in equation (3.100) require
simplification, so that it can be expressed in terms of the three primary variables; wu , au
and T . The simplification of each term is as following (Ramesh, 1996):
1- The time derivative of heat capacity which is represented by the second term in equation
(3.100) can be written as:
tSnC
tnSCSCSCC
tH w
wpwdaapdavapvwwpwspsc )(
t
SnCt
SnCt
SnCt
SnC daapda
adapda
vapv
avpv (3.101)
Equation (3.101) can be reduced to
tSnC
tSnC
tSn
tn
tH da
apdav
apvw
cpcpc
21 (3.102)
where,
daapdavapvwwpwspscp SCSCSCC1 (3.103)
dapdavpvwwpwcp CCSC2 (3.104)
Substituting the degree of saturation from equation (3.21), the vapour density derivatives
from equation (3.35) and the dry air density from equation (3.80) into equation (3.102)
yields:
tuC
tTC
tuCSnC
tuK
tTK
tuKSnC
tu
sS
tT
TS
tu
sSn
tH
adaadaT
wdawapda
afafT
wfwapv
awwwwcp
c2
or,
tuC
tTC
tuC
tH a
TaTTw
Twc
111 (3.105)
Chapter (3) Governing equations of coupled fluid flow and solute transport
84
where,
dawapdafwapvw
cpTw CSnCKSnCs
SnC 21 (3.106)
daTapdafTapvw
cpTT CSnCKSnCT
SnC 21 (3.107)
daaapdafaapvw
cpTa CSnCKSnCs
SnC 21 (3.108)
2- The time derivative of degree of saturation is represented by the third term in equation
(3.100). Substituting equations (3.7) and (3.21) the third term can be written as:
tSLn
tSLn w
va
v)1(
tuC
tTC
tuC
tSLn a
TaTTw
Twa
v 222 (3.109)
where,
sSLnC w
vTw2 (3.110)
TSLnC w
vTT 2 (3.111)
sSLnC w
vTa2 (3.112)
3- Similarly, the time derivative of vapour density is represented by the forth term in
equation (3.100). Substituting equations (3.35) the forth term can be written as:
tuK
tTK
tuKLnS
tLnS a
fafTw
fwav
a
tLnS v
a = t
uCtTC
tuC a
TaTTw
Tw 333 (3.113)
Chapter (3) Governing equations of coupled fluid flow and solute transport
85
where,
fwaTw KLnSC 3 (3.114)
fTaTT KLnSC 3 (3.115)
faaTa KLnSC 3 (3.116)
4- Substituting the gradients of vapour velocity from equation (3.46) into the sixth term in
equation (3.100) yields:
tuK
tTK
tuKL
tvL a
vavTw
vwwv
w
tvL v
w tuK
tTK
tuK a
TaTTw
Tw 111 (3.117)
where,
vwwTw KLK 1 (3.118)
vTwTT KLK 1 (3.119)
vawTa KLK 1 (3.120)
5- The seventh term of equation (3.100) can be extended as:
)()( vaavva vvLvL
Substituting equations (3.22) and (3.35) into this term, will result:
tuK
tTK
tuKvuKLvL a
fafTw
fwaaavva )( (3.121)
Finally by substituting equations (3.105), (3.109), (3.113), (3.117) and (3.121) into (3.100),
the equation (3.100) can be written as:
Chapter (3) Governing equations of coupled fluid flow and solute transport
86
)()(
)()(
)()(
)(
111
333222
111
radapdaravpv
rvwpvrwwpw
ava
TaTTw
TwT
aTaTT
wTw
aTaTT
wTw
aTaTT
wTwrc
TTvCTTvC
TTvCTTvC
vLt
uKtTK
tuKT
tuC
tTC
tuC
tuC
tTC
tuC
tuC
tTC
tuCTT
tTH
(3.122)
More simplification for the last five terms of equation (3.122) can be done as:
)()( radapdaavpvvwpvwwpwav TTvCvCvCvCvL
adadaapdaravvapvr
vwpvrwwpwrav
vvCTTvvCTT
vCTTvCTTvL
)()(
)()()( (3.123)
By substituting the equations for the velocities of water, air and vapour (3.9), (3.22), (3.46)
and equations for the gradients for the densities of vapour and dry air (3.35), (3.80) into
equation (3.123) the equation will be (Ramesh, 1996):
)()(222
zCTTuVTVuVuKTKuK
wpwraTaTT
wTwaTaTTwTw (3.124)
where,
vwpvw
wwpwwrTw KC
KCTTK )(2 (3.125)
vTpvwrTT KCTTK )(2 (3.126)
avapdadaapvvvapvwrTa KLKCKCKCTTK ))((2 (3.127)
fwadawapdafwapvrTw KLvCvCKvCTTV ))(( (3.128)
fTadaTpdafTpvar
daapdavapvwvpvwwpwTT
KLvCCKCvTT
TvCvCvCvCV
)()(
)( (3.129)
faadaapdafapvarTa KLvCCKCvTTV )()( (3.130)
Chapter (3) Governing equations of coupled fluid flow and solute transport
87
Substituting equation (3.124) into (3.122) yields the governing equation for heat transfer in
terms of the three primary variables; pore water pressure wu , pore air pressure au and
absolute temperature T as following:
TaTaTTwTw
aTaTTwTwa
TaTTw
Tw
JuVTVuV
uKTKuKt
uCtTC
tuC (3.131)
where,
32)( TwTwrTwTw CCTTCC (3.132)
321 )( TTTTrTTcTT CCTTCHC (3.133)
321 )( TaTarTaTa CCTTCC (3.134)
21 TwTwTw KKK (3.135)
21 TTTTTTT KKK (3.136)
21 TaTaTa KKK (3.137)
)()( zKCTTJ wwPwrT (3.138)
3.6 The governing equation for solute transport This section is focusing on the phenomena that effect solute transport in soils. Several
studies have evaluated various aspects of solute transport processes in saturated and
unsaturated soils. Many analytical and numerical solutions have been developed for
predicting the flow of solutes in groundwater. The majority of the work that has been done
for unsaturated flow assumed constant or uniform water content. However, there has been
relatively little work done for solute transport in a transient water and air flow field (Sander
and Braddock, 2005). To study solute transport mechanisms consider a unit volume of soil
(Figure 3.1). This volume is called a control volume and the boundaries of the element are
called control surfaces (Istok, 1989).
Chapter (3) Governing equations of coupled fluid flow and solute transport
88
Figure 3.1 Control volume of solute transport through pours media
The law of conversation of mass for solute transport requires that the rate of change of
solute mass within the control volume is equal to the net rate at which solute is entered to
the control volume through the control surface plus the net rate at which solute is produced
within the control volume by various chemical and physical processes.
3.6.1 Net rate of solute inflow the control volume
The net rate of solute inflow the control volume is the difference between inflow and
outflow and can be written as:
Net rate of solute inflow = ))((
))(())((
zzzz
yyyyxxxx
FFF
FFFFFF
Net rate of solute inflow )()()( zzyyxx FFF (3.139)
In porous media, solute transport occurs mainly by various processes including advection,
diffusion, dispersion, adsorption, radioactive decay and biological degradation. These
processes will be described in details (Istok, 1989).
Z
)( yyy FF
Y
X
)( zzz FF
Fz
Fy
Fx
)( XxX FF X
YZ
Chapter (3) Governing equations of coupled fluid flow and solute transport
89
3.6.1.1 Advection
The process by which solutes are transported by the bulk motion of the flowing
groundwater is called advection. The rate of solute transport that occurs by advection is
given by the product of the solute concentration c and the components of apparent
groundwater velocity xv , yv . In terms of the two components of the solute transport in the
x and y directions, the rate of solute transported by advection can be written as:
cvFxwadvectionx, (3.140)
cvFywadvectiony , (3.141)
3.6.1.2 Diffusion
The process by which solutes are transported by the random thermal motion of solute
molecules is called diffusion. The rate of solute transport that occurs by diffusion is given
by Fick’s Law. In terms of the two components of the solute transport in the x and y
directions, the rate of solute transported by diffusion can be written as:
xcDF mdiffusionx, (3.142)
ycDF mdiffusiony , (3.143)
Where mD is the molecular diffusion coefficient in the porous medium [L]2[T]-1.
The molecular diffusion coefficient for a solute in porous media is much smaller than the
molecular diffusion coefficient for the same solute in aqueous solution. An empirical
relationship for the molecular diffusion coefficient mD can be written as (Istok, 1989):
mom DD )( (3.144)
Chapter (3) Governing equations of coupled fluid flow and solute transport
90
where,
moD : is the molecular diffusion coefficient in aqueous solution [L]2[T]-1
: is the volumetric water content
: is an empirical correction factor (ranging from 0.01 for very dry soils to 0.5 for
saturated soils).
Values of the molecular diffusion coefficient are in the range of 1.E-8 to 1.E-10 [L]2[T]-1 at
25o C. Molecular diffusion coefficients are strongly temperature dependent, but are only
weakly dependant on the concentration of other dissolved species. The small size of
molecular diffusion coefficient means that the rate of solute transport by diffusion is
usually very small relative to the rate of solute transport by advection and dispersion and
may be negligible (Istok, 1989).
3.6.1.3 Mechanical Dispersion
Mechanical dispersion (or hydraulic dispersion) is a mixing or spreading process caused by
small-scale fluctuation in groundwater velocity along the tortuous flow paths within
individual pores. On a much larger scale mechanical dispersion can also be caused by the
presence of heterogeneities within the aquifer. The rate of solute transport by mechanical
dispersion is given by a generalized form of Fick’s Law of diffusion. In terms of the two
components of solute transport in x and y directions, the rate of solute transport by
mechanical dispersion can be written as:
)()(, cy
Dcx
DF wwwwdispersionx xyxx (3.145)
)()(, cy
Dcx
DF wwwwdispersiony yyyx (3.146)
where xxwD ,
xywD , yxwD ,
yywD are the coefficients of dispersivity tensor [L][T]-1.
For water phase the coefficients of mechanical dispersion can be computed from the
following expressions (Bear, 1987):
Chapter (3) Governing equations of coupled fluid flow and solute transport
91
wy
wx
wxx mw
Tw
Lw Dv
v
v
vD
22
(3.147)
w
y
w
x
wyy mw
Lw
Tw Dv
vv
vD
22
(3.148)
wyx
wwyxxy mww
TLww Dv
vvDD )( (3.149)
where,
wT : is the transverse dispersivity for water phase [L].
Lw : is the longitudinal dispersivity for water phase [L].
wmD : is the coefficient of water molecular diffusion [L]2[T]-1
xwv : is the component of the pore water velocity in x direction w
xw
vvx
ywv : is the component of the pore water velocity in y direction w
yw
vv
y
v : is the magnitude of the water velocity ( 22yx ww vvv )
By substituting from equations (3.140), (3.141), (3.145) and (3.146) for the values of
advection, dispersion and diffusion into equation (3.139) the net ret of solute inflow can be
written as:
The net ret of solute inflow =
))(
)()(
)()(
Cvy
Cvx
cy
Dcx
Dy
cy
Dcx
Dx
yx
yyyx
xyxx
ww
wwww
wwww
(3.150)
Chapter (3) Governing equations of coupled fluid flow and solute transport
92
3.6.2 Net rate of solute production within the control volume
Several processes can act as sources or sinks for solute transport within the control volume
including sorption/desorption, chemical reaction or biological degradation. For the case of
transport involving a sorption/desorption reaction (Istok, 1989):
AA (3.151)
The net rate of reaction r between dissolved species A and sorbed species A , can be
written as:
tC
btCr (3.152)
where,
: is the volumetric water content of the porous medium.
b : is the bulk density of the porous medium.
C : is the concentration of the dissolved species A (mass of solute / volume of
groundwater).
C : is the concentration of sorbed species A (mass of solute / volume of dry porous
media).
Equation (3.152) can be written as:
CkCkr rf (3.153)
where,
fk : is the constant for the forward reaction )( AA and
rk : is the constant for reverse reaction )( AA .
Chapter (3) Governing equations of coupled fluid flow and solute transport
93
If the net rate of reaction is assumed to be zero (e.g., the reaction is in equilibrium),
equation (3.153) can be solved directly for the concentration of the sorbed species A and
written as:
CKCC dkk
r
f (3.154)
Where dK is the equilibrium distribution coefficient [L]3[M]-1.
The net rate of solute production due to a sorption/desorption reaction between a solute and
the porous medium within the control volume can be obtained by combining equations
(3.152) and (3.154), and be written as:
tC
dbsorptiontC K))( (3.155)
If the solute also undergoes radioactive decay or biological degradation, the net rate of
solute production by this mechanism can be written as:
)())( CKC dbdecaytC (3.156)
where,
: is the decay constant for the solute.
The net rate of solute production due to a sorption/desorption reaction, radioactive decay or
biological degradation between a solute and the porous medium within the control volume
can be obtained by combining equations (3.155) and (3.156) and be written as:
The net ret of solute production = )( CKCK dbtC
db (3.157)
Chapter (3) Governing equations of coupled fluid flow and solute transport
94
3.6.3 Governing equation of solute transport in saturated soil
By substituting equations (3.150) and (3.157) for the values the net rate of solute inflow
and the net rate of solute production the governing equation of solute transport in saturated
soil can be written as:
)()()()()( cy
Dcx
Dy
cy
Dcx
Dxt
cwwwwwwww
wwyyyxxyxx
)()())( cKccKcvy
cvx dbdbtww yx
(3.158)
Equation (3.158) can be written using simpler notations as:
0)()()()()(wdbwwwwwwwdb
ww cKccDcvcKtt
c (3.159)
Equation (3.159) can be modified to include a retardation factor R as (AL-Najjar, 2006):
0)()()( RccDcvt
cRwwwwwwww
ww (4.160)
where,
w
dbKR 1 (3.161)
Equation (3.160) can be written using simpler notations as (Javadi et al., 2008):
0)()()( ccDvctc (3.162)
where,
wR
wvv
Chapter (3) Governing equations of coupled fluid flow and solute transport
95
wwDD
Rww
wcc
3.6.4 Governing equation of solute transport in unsaturated soil
Modelling the solute transport in the water and air phases requires a description of the mass
transfer processes between the two phases. The concentration in each phase is proportional
to the concentration in the other phase and they are related by Henry’s Law:
w
accH (3.163)
where, H : is Henry’s constant [T]2[L] -1[M]-1.
For modelling the fluid flow and solute transport in unsaturated soils it is considered in this
work that non-uniform velocity of water wv , velocity of air av , volumetric water content
w and volumetric air content a , will change with time according to the changes in the
pore water pressure wu , pore air pressure au , water degree of saturation wS and air degree
of saturation aS (AL-Najjar, 2006).
The rate of solute transport by advection will include the effect of the air phase, so equation
(3.140) and (3.141) can be written as:
))((, cHvvFxx awadvectionx (3.164)
))((, cHvvFyy awadvectiony (3.165)
Also, the rate of solute transport by mechanical dispersion will include the effect of the air
phase, so equation (3.145) and (3.146) can be written as:
Chapter (3) Governing equations of coupled fluid flow and solute transport
96
)()()()(, cy
Dcx
Dcy
Dcx
DF aaaawwwwdispersionx xyxxxyxx (3.166)
)()()()(, cy
Dcx
Dcy
Dcx
DF aaaawwwwdispersiony yyyxyyyx (3.167)
where xxaD ,
xyaD , yxaD ,
yyaD are the coefficients of dispersivity tensor [L][T]-1 for air
phase, the coefficients of mechanical dispersion can be computed from the following
expressions:
ayx
axx ma
Taa
La Dv
v
v
vD
22
(3.168)
a
y
a
x
ayy ma
La
Ta Dv
vv
vD
22
(3.169)
ayx
ayxxy maa
TaLaa Dv
vvDD )( (3.170)
aT : is the transverse dispersivity for air phase [L].
aL : is the longitudinal dispersivity for air phase [L].
xav : is the component of the pore air velocity in x direction a
xa
vvx
yav : is the component of the pore air velocity in y direction a
ya
vv y
v : is the magnitude of the air velocity ( 22yx aa vvv )
amD : is the coefficient of air molecular diffusion [L]2[T]-1.
The governing equation of solute transport for unsaturated soil can be written similar to
equation (3.158) as:
Chapter (3) Governing equations of coupled fluid flow and solute transport
97
0)/1)(()(
)()()()(
)()()()(
)()()()())((
cKHCK
cx
Dcy
Dy
cycDc
xD
xH
cx
Dcy
Dy
cy
Dcx
Dx
cvy
cvx
cvy
cvxt
cH
wdbaawwdbt
aaaaaaaa
wwwwwwww
aawwwaw
yxyyxyxx
yxyyxyxx
yxyx
(3.171)
where a is the chemical reaction rate for air [T]-1.
Equation (3.171) can be written using simpler notations similar to equation (3.159) as
following:
0)/1)((
)()()()()())((
cKH
cDcDcvcvCKt
cH
wdbaaww
aawwawdbtwaw
(3.172)
Equation (3.172) can be modified to include a retardation factor R as:
0)(
)()()()())((
RcH
cDcDcvcvt
cHR
aaww
aawwawwaw
(3.173)
Equation (3.173) can be written using simpler notations as (Javadi et al., 2008):
0)()())(( ccDvct
cHR aw (3.174)
or
0)()()( ccDvctc (3.175)
where,
Chapter (3) Governing equations of coupled fluid flow and solute transport
98
)()(yxyx aaww vvHvvv
yxyyxyxxyxyyxyxx aawaawwwww DDDDHDDDDD
RH aaww )(
)( aw HR
3.7 The governing differential equations
Equations (3.60), (3.88), (3.131) and (3.175) are considered as the governing deferential
equations for the coupled flow and transport in unsaturated soils and are listed below:
Moisture transfer:
tuC
tTC
tuC a
wawTw
ww = wawawtwww JuKTKuK )()()( (3.176)
Air transfer:
tuC
tTC
tuC a
aaaTw
aw = aaaawaw JuKuK )()( (3.177)
Heat transfer:
TaTaTTwTw
aTaTTwTwa
TaTTw
Tw
JuVTVuV
uKTKuKt
uCtTC
tuC (3.178)
Solute transport:
0)()()( ccDvctc (3.179)
The above four equations define the complete formulation of coupled flow and solute
transport in unsaturated soil. The transient flow equations are solved in terms of three
variables; pore water pressure wu , pore air pressure au and absolute temperature T at
different times. The transient solute transport equation is then solved for the value of solute
concentration c at different times. This procedure is valid when changes in groundwater
density due to changing solute concentration can be assumed to be small (Istok, 1998).
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
99
Numerical solution of governing equations of coupled fluid flow
and solute transport
4.1 Introduction
The governing differential equations of fluid flow and solute transport have been defined in
chapter 3. The coupled fluid flow and solute transport in unsaturated soil is modeled using
two sets of equations. The first set of equations describes water, air and heat flow. The
second set describes solute transport. The numerical approach to achieve a solution to the
governing differential equations will be described in this chapter. The nonlinear governing
differential equations are solved using a finite element method in the space domain and a
finite difference scheme in the time domain. The solution is divided into two parts. In the
first part the flow equations for water, air and heat transfer will be solved simultaneously
for the three primary variables; pore water pressure ( wu ), pore air pressure ( au ) and
absolute temperature (T ). In the second part solute transport equation will be solved for
solute concentration (c). This chapter presents an overview of the different mathematical
models that can be used to solve coupled fluid flow and solute transport problem focusing
on finite element and finite difference methods which are used in this work and then the
numerical solution of the governing differential equations of the coupled fluid flow and
solute transport is presented.
4.2 Solution methods for differential equations
Solution of differential equations governing a physical phenomenon means finding the
unknowns in the governing equations with the associated boundary and initial conditions.
There are a number of methods that can be used to solve such problems. The wide spectrum
of these methods can be classified as analytical and numerical methods. Figure 4.1 shows a
summary of solution methods for differential equations. These methods are described in the
following sections with focous on the methods which are used in this study.
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
100
Figure 4.1 Summary of the solution methods for differential equations
4.2.1 Analytical methods
Analytical methods can be used to obtain exact solutions for a number of problems. They
are particularly used for linear problems and regions of simple or regular geometry. For
nonlinear problems with regions of irregular geometry, very few analytical solutions exist
and these are usually approximate solutions. The accuracy of the analytical methods can be
Solution methods of
differential equations
Analytical methods Numerical methods
Finite difference
method
Method of characteristics
Finite element method
Finite volume method
1- Separation of variable 2- Similarity solutions 3- Complex variable
technique 4- Fourier and Laplace
transformation 5- Green function 6- Regular and singular
perturbations 7- Power series
Weighted residual method
Virtual work method
Subdomain collocation
method
Galerkin method
Point collocation
method
Least squares method
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
101
very good and exact in some cases but it requires several assumptions to simplify the
problem which may be not valid for complicated problems. Many physical phenomena
such as, coupled fluid flow and solute transport in porous media are governed by complex
nonlinear partial differential equations. Solving such complex problems of nonlinear partial
differential equations is generally intractable analytically. For such problems, the use of
numerical methods to obtain approximate solutions is advantageous and preferable.
4.2.2 Numerical methods
A number of numerical methods can be used to solve complex nonlinear partial differential
equations (Figure 4.1). The finite difference and finite element methods are the most
popular methods that have been applied in the field of coupled fluid flow and solute
transport in porous media.
4.2.2.1 Finite difference method
Finite difference method (FDM) is popular in simulation of large systems. The reason for
this popularity is that the finite difference method is conceptually straightforward and the
fundamental concepts are simply understood. The finite difference method is most suitable
for problems with simple or regular geometry. It can be used in simple problems (e.g.,
isotropic and homogenous aquifer with regular boundary). The accuracy of finite difference
method for such simple problems is quite good but this method becomes complicated and
inefficient when the problem is nonlinear with irregular domain (i.e., anisotropic and
heterogeneous aquifer with irregular boundary), (Istok, 1989).
4.2.2.2 Finite element method
The finite element method (FEM) has become a powerful numerical tool for the solution of
a wide range of engineering problems. The finite element method was first used to solve
groundwater flow and solute transport problems in the early 1970’s (Istok, 1989). Since
then, the method has gained wide acceptance in the field of fluid flow and solute transport
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
102
modeling. The finite element method is a technique which can produce near optimal
approximate solutions to the partial differential equations of a certain problem. The
implementation ultimately employs subdivision of the problem domain into a number of
elements with small regions called finite elements. Many shapes are available for elements.
The fundamental idea of the finite element method is to produce a set of algebraic
equations, written on each finite element which are then collected together using a
procedure called assembly to form the global matrix statement. The approximate values of
the dependent variables are then determined via solution of this matrix statement using any
linear algebra technique (Baker and Pepper, 1991).
The finite element method has a number of advantages that made it very popular (Daryl,
2002). The advantages of this method include the ability to handle;
1. irregular and curved geometric boundaries,
2. variable spacing of the nodes,
3. variable size of the elements,
4. unlimited numbers and kinds of boundary conditions,
5. anisotropic and heterogeneous materials,
6. nonlinear behavior and
7. dynamic effects.
The finite element method is useful in virtually every field of engineering analysis (Klaus,
1996). In this method, a complex region defining a continuum is discretized into simple
geometric shapes called finite elements. The material properties and governing
relationships are considered over these elements and expressed in terms of unknown values
at element nodes. An assembly process, results in a set of equations. Solution of these
equations gives the approximate behaviour of the continuum (Chandrupatla and Belegundu,
1991).
Certain steps in formulating a finite element model of solute transport problems are
embodied in the developed finite element model. The steps of finite element analysis can be
summarized as follows (Cheung et al., 1996):
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
103
1. Discretization of the problem domain into a number of sub-regions known as finite
elements.
2. Selection of node and element interpolation functions.
3. Evaluation of individual element properties.
4. Formation of elements stiffness matrices.
5. Assembly of element matrices to form the global matrix.
6. Definition of the physical constraints (boundary and initial conditions).
7. Formulation of the global unknown vector.
8. Solution of system of equations for the unknown nodal variables.
9. Finally, computation of further parameters such as, hydrostatic head, fluid density,
velocity, dispersion coefficients and other physically meaningful quantities from the
computed nodal variables and element properties .
4.3 Numerical solution of the governing equations
The numerical solution of governing differential equations of coupled fluid flow and solute
transport will be described in this section. The current work employs a two dimensional
eight noded isoparametric element. The spatial discretization of the differential equations
using the finite element method will be described in details. The solution method of
temporal integration using the finite difference method is then employed to obtain the
transient solution of the spatially discretised equations for both fluid flow and solute
transport. The flow equations for water, air and heat transfer will be solved simultaneously
to determine pore water pressure ( wu ), pore air pressure ( au ) and absolute temperature
(T ), and then solute transport equation will be solved for solute concentration c. The
coupling mechanism of fluid flow and solute transport is presented in the final part of this
chapter.
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
104
4.3.1 Spatial discretization of the moisture transfer governing equation
The governing differential equation for moisture transfer was derived in chapter 3, in terms
of three primary unknowns; wu , au and T. Equation (3.60) may be rearranged and rewritten
as:
0][][][t
uCtTC
tuCJuKTKuK a
wawTw
wwwawawTwww (4.1)
The primary unknowns can be approximated using the shape function approach for eight
node isoparametric element as:
wssww uNuu8
1ˆ (4.2)
ssTNTT8
1
ˆ (4.3)
assaa uNuu8
1ˆ (4.4)
where,
sN : is the shape function.
wsu : is the nodal pore water pressure.
sT : is the nodal temperature.
asu : is the nodal pore air pressure.
The derivatives of the shape functions for equations (4.2) (4.3) and (4.4) can be expressed
as:
wss
sw uNu ).(ˆ8
1 (4.5)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
105
ss
s TNT ).(ˆ8
1 (4.6)
ass
sa uNu ).(ˆ8
1 (4.7)
By replacing the primary unknowns in equation (4.1) with their shape functions
approximations from equation (4.2), (4.3) and (4.4), equation (4.1) can be written as:
wa
wawTw
wwwawawTwww Rt
uCtTC
tuCJuKTKuK
ˆˆˆ]ˆ[]ˆ[]ˆ[ (4.8)
where,
wR : is the residual error introduced by the approximation functions.
Employing the Galerkin weighted residual approach to minimize the residual error,
equation (4.8) can be written as (Ramesh, 1996):
0ˆˆˆ
]ˆ[]ˆ[]ˆ[ e
e
awawT
wwwwawawTwwwr d
tuC
tTC
tuCJuKTKuKN (4.9)
where,
e : is the element domain.
rN : is the shape function.
Employing integration by parts to the first term of equation (4.9) yields:
e
erwww
e
ewwwr
e
ewwwr dNuKduKNduKN ˆˆˆ (4.10)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
106
Similarly, the second and third terms of the equation (4.9) can be written as:
e
erwT
e
ewTr
e
ewTr dNTKdTKNdTKN ˆˆˆ (4.11)
e
erawa
e
eawar
e
eawar dNuKduKNduKN ˆˆˆ (4.12)
The fourth term of the equation (4.9) has been defined in chapter 3 as:
)( zKJ www
Introducing this substitution and employing integration by parts, the fourth term can be
written as:
e
ewwr
e
ewr dzKNdJN )(
e
erww
e
ewwr dNzKdzKN )()( (4.13)
Substituting equations (4.10), (4.11), (4.12) and (4.13) into equation (4.9), will result:
0ˆˆˆ
)()(ˆˆ
ˆˆˆˆ
eawawT
wwwr
rwwwwrrawaawar
erwTwTrrwwwwwwr
dt
uCtTC
tuCN
NzKzKNNuKuKN
NTKTKNNuKuKN
(4.14)
Equation (4.14) can be rearranged and rewritten as:
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
107
0ˆˆˆ
)(ˆˆˆ
)(ˆˆˆ
eawawT
wwwr
erww
erawa
erwT
erwww
e
ewwr
eawar
ewTr
ewwwr
dt
uCtTC
tuCN
dNzKdNuKdNTKdNuK
dzKNduKNdTKNduKN
(4.15)
Employing the Gauss-Green divergence theorem into the first term of equation (4.15)
yields (Ramesh, 1996):
e
ewwawawTwwwr
e
ewwawawTwwwr
dnzKuKTKuKN
dzKuKTKuKN
..ˆˆˆ
.ˆˆˆ
(4.16)
where,
e : is the element boundary surface.
Substituting equation (4.16) into equation (4.15) yields:
0..ˆˆˆ
ˆˆˆ
)(ˆˆˆ
e
ewwawawTwwwr
eawawT
wwwr
e
erww
erawa
erwT
erwww
dnzKuKTKuKN
dt
uCtTC
tuCN
dNzKdNuKdNTKdNuK
(4.17)
The surface integral in equation (4.17) is equal to the sum of liquid and vapor fluxes normal
to the boundary surface. Substituting equations (3.9), (3.22) and (3.46) for water, air and
vapour velocities into the last term of equation (4.17) yields:
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
108
e
evawvdwwnwr
e
eaawvTavawvww
w
wwwr
e
ewwawawTwwwr
dvvvN
dnuKTKuKuKzu
KN
dnzKuKTKuKN
.ˆˆˆ
..ˆˆˆˆˆ
..ˆˆˆ
(4.18)
where,
wnv̂ : is the approximation of water velocity normal to the boundary surface.
vdv̂ : is the approximation of diffusive vapour velocity normal to the boundary surface.
vav̂ : is the approximation of pressure vapour velocity normal to the boundary surface.
By introducing the derivatives of the shape functions from equations (4.5), (4.6) and (4.7)
into equation (4.17) and substitute from equation (4.18), the final form of equation (4.17)
can be written as:
0.ˆˆˆ
)()()()(
))(())(()))((
e
evawvdwwnwr
easswa
sswT
wsswwr
erww
e
erasswa
ersswT
erwssww
dvvvN
dtuNC
tTNC
tuNCNdNzK
dNuNKdNTNKdNuNK
0.ˆˆˆe
evawvdwwnw
Tase
e
Twa
Se
e
TwT
wse
e
Tww
e
e
Tww
ase
e
Twas
e
e
TwTws
e
e
Tww
dvvvNt
udNNC
tTdNNC
tudNNCdzNK
udNNKTdNNKudNNK
(4.19)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
109
where,
N : is the shape function matrix.
The spatial discretization of the governing differential equation for water flow can then be
written as:
wwawaswTwswwas
waS
wTws
ww fuKTKuKt
uCt
TCt
uC (4.20)
where,
em
e eww
Tww dNCNC
1 (4.21)
em
e ewT
TwT dNCNC
1 (4.22)
em
e ewa
Twa dNCNC
1 (4.23)
em
e eww
Tww dNKNK
1 (4.24)
em
e ewT
TwT dNKNK
1 (4.25)
em
e ewa
Twa dNKNK
1 (4.26)
m
e e
evawvdwwnw
Tem
e eww
Tw dvvvNdzKNf
11.ˆˆˆ)( (4.27)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
110
4.3.2 Spatial discretization of the air transfer governing equation
The governing differential equation for air transfer was also derived in chapter 3, in terms
of three primary unknowns; wu , au and T. Equation (3.88) may be rearranged and rewritten
as:
0)(t
uCtTC
tuCJuKuK a
aaaTw
awaaaawaw (4.28)
By replacing the primary unknowns in equation (4.28) with their shape functions
approximations from equation (4.2), (4.3) and (4.4), equation (4.28) can be written as:
aa
aaaTw
awaaaawaw Rt
uCtTC
tuCJuKuK
ˆˆˆ)ˆ(ˆ (4.29)
where,
aR : is the residual error introduced by the approximation functions.
Employing Galerkin weighted residual approach to minimize the residual error, equation
(4.29) may be written as (Ramesh, 1996):
0ˆˆˆ
)ˆ(ˆ e
e
aaaaT
wawaaaawawr d
tuC
tTC
tuCJuKuKN (4.30)
Employing integration by parts to the first term of equation (4.30) yields:
e
erwaw
e
ewawr
e
ewawr dNuKduKNduKN ˆˆˆ (4.31)
Similarly, the second term of the equation (4.30) can be written as:
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
111
e
eraaa
e
eaaar
e
eaaar dNuKduKNduKN ˆˆˆ (4.32)
The third term of the equation (4.30) has been defined in chapter 3 as:
)( zKHJ wdaaa
Introducing this substitution and employing integration by parts, the third term can be
written as:
e
ewadar
e
ear dzKHNdJN )(
e
eradaw
e
ewadar dNzHKdzKHN )( (4.33)
Substituting equations (4.31), (4.32) and (4.33) into equation (4.30) leads to:
0ˆˆˆ
)(
ˆˆˆˆ
eaaaaT
wawrradawwadar
eraaaaaarrwawwawr
dt
uCtTC
tuCNNzHKzKHN
NuKuKNNuKuKN
(4.34)
Equation (4.34) can be rearranged and rewritten as:
0ˆˆˆ
ˆˆ
)(ˆˆ
eaaaaT
wawr
radawraaarwaw
ewadaraaarwawr
dt
uCtTC
tuCN
NzHKNuKNuK
zKHNuKNuKN
(4.35)
Employing Gauss-Green divergence theorem to the first term of equation (4.35) yields :
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
112
e
ewadaaaawawr
e
ewadaaaawawr
dnzKHuKuKN
dzKHuKuKN
.)(ˆˆ
)(ˆˆ
(4.36)
Substitution of equation (4.36) into (4.35) results:
0.)(ˆˆ
ˆˆˆ
ˆˆ
e
ewadaaaawawr
eaaaaT
wawr
eradawraaarwaw
dnzKHuKuKN
dt
uCtTC
tuCN
NzHKNuKNuK
(4.37)
The surface integral in equation (4.37) is equal to the sum of free and dissolved air fluxes
normal to the boundary surface and can be written as (Ramesh, 1996):
e
eadawadar
e
edawaar
e
ewadaaaawawr
dnvvHN
dnvHvN
dnzKHuKuKN
..)ˆ()ˆ(
..)ˆˆ(
.)(ˆˆ
(4.38)
Substitution for the water and air velocities defined in chapter 3 in equations (3.9) and
(3.22) into equation (4.38) leads to:
e
eandafndar
e
eaada
w
wwadar
dvvN
dnuKzuKHN
.ˆˆ
.. (4.39)
where,
fnv̂ : is the approximation of velocity of free dry air.
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
113
anv̂ : is the approximation of velocity of dissolved dry air.
By introducing the derivatives of the shape functions from equations (4.5), (4.6) and (4.7)
into equation (4.37) and substituting from equation (4.39), the final form of equation (4.37)
can be written as:
0.ˆˆee
ee
e daee
eanfnda
TaseTaa
SeTaT
wseTaw
eTawas
eTaaws
eTaw
dvvNt
udNNC
tTdNNC
tudNNC
dzNHKudNNKudNNK
(4.40)
The spatial discretization of governing differential equation for air flow can be written as:
aasaawsawas
aas
aTws
aw fuKuKt
uCt
TCt
uC (4.41)
where,
em
e eaw
Taw dNCNC
1 (4.42)
em
e eaT
TaT dNCNC
1 (4.43)
em
e eaa
Taa dNCNC
1 (4.44)
em
e eaw
Taw dNKNK
1 (4.45)
em
e eaa
Taa dNKNK
1 (4.46)
m
e e
eanfnda
Tem
e eadaw
Ta dvvNdzHKNf
11.ˆˆ)( (4.47)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
114
4.3.3 Spatial discretization of the heat transfer governing equation
The governing differential equation for heat transfer was also derived in chapter 3, in terms
of three primary unknowns; wu , au and T. Equation (3.131) may be rearranged and
rewritten as:
0t
uCtTC
tuCJuVTVuV
uKTKuK
aTaTT
wTwTaTaTTwTw
aTaTTwTw
(4.48)
By replacing the primary unknowns in equation (4.48) with their shape functions
approximations from equation (4.2), (4.3) and (4.4), equation (4.48) can be written as:
Ta
TaTTw
TwTaTaTTwTw
aTaTTwTw
Rt
uCtTC
tuCJuVTVuV
uKTKuK
ˆˆˆˆˆˆ
ˆˆˆ (4.49)
where,
TR : is the residual error introduced by the approximation functions.
Employing Galerkin weighted residual approach to minimize the residual error, equation
(4.49) may be written as (Ramesh, 1996):
0ˆˆˆˆˆˆ
ˆˆˆ
eaTaTT
wTwTaTaTTwTw
aTaTTwTwr
dt
uCtTC
tuCJuVTVuV
uKTKuKNe
(4.50)
Employing integration by parts to the first term of equation (4.50) yields:
e
erwTw
e
ewTwr
e
ewTwr dNuKduKNduKN ˆˆˆ (4.51)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
115
Similarly, the second and the third terms of the equation (4.50) can be written as:
e
erTT
e
eTTr
e
eTTr dNTKdTKNdTKN ˆˆˆ (4.52)
e
eraTa
e
eaTar
e
eaTar dNuKduKNduKN ˆˆˆ (4.53)
The seventh term of the equation (4.50) has been determined in chapter 3 as:
)()( zKCTTJ wwPwrT
Introducing this substitution and employing integration by parts, the seventh term can be
written as:
e
ewwPwrr
e
eTr dzKCTTNdJN )()(
e
erwwPwr
e
ewwPwrr dNzKCTTdzKCTTN )()( (4.54)
Substituting equations (4.51), (4.52), (4.53) and (4.54) into equation (4.50) leads to:
0ˆˆˆ
)()(
ˆ
ˆˆˆˆ
ˆˆˆˆ
e
e
aTaTT
wTwr
e
erwwPwr
e
ewwPwrr
e
eaTar
e
eTTr
e
ewTwr
e
eraTa
e
eaTar
e
erTT
e
eTTr
e
erwTw
e
ewTwr
dt
uCtTC
tuCN
dNzKCTTdzKCTTN
duVN
dTVNduVNdNuKduKN
dNTKdTKNdNuKduKN
(4.55)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
116
Equation (4.55) can be rearranged as:
0ˆˆˆ
)(ˆˆˆ
ˆˆˆ
)(ˆˆˆ
eaTaTT
wTwr
erwwPwr
eraTa
erTT
erwTw
eaTar
eTTr
ewTwr
ewwPwrr
eaTar
eTTr
ewTwr
dt
uCtTC
tuCN
dNzKCTTdNuKdNTKdNuK
duVNdTVNduVN
dzKCTTNduKNdTKNduKN
e
eeee
eee
eeee
(4.56)
Applying Gauss-Green divergence theorem for the first term of equation (4.56) yields:
e
ewwPwraTaaTaTTwTwr
e
ewwPwraTaaTaTTwTwr
dnzKCTTuKuKTKuKN
dzKCTTuKuKTKuKN
..)(ˆˆˆˆ
)(ˆˆˆˆ
(4.57)
Substituting (4.57) into equation (4.56) results:
0..)(ˆˆˆˆ
ˆˆˆ_
ˆˆˆ
)(
ˆˆˆ
ewwPwraTaaTaTTwTwr
eaTar
eTTr
ewTwr
eaTaTT
wTwr
erwwPwr
eraTa
erTT
erwTw
dnzKCTTuKuKTKuKN
duVNdTVNduVN
dt
uCtTC
tuCN
dNzKCTT
dNuKdNTKdNuK
e
eee
e
e
eee
(4.58)
The surface integral in equation (4.58) is equal to the sum of free and dissolved heat fluxes
normal to the boundary surface. Substituting equations (3.9), (3.22) and (3.46) for water, air
and vapour velocities into the last term of equation (4.58) yields:
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
117
e
ehhhhhhr
eaadapdaraavpvr
vTwvwavawpvr
wwpwrwwwpwr
aave
vTwvwavawTr
e
ewwPwraTaaTaTTwTwr
dFFFFFFN
dnuKCTTuKCTT
TKuKuKCTT
zKCTTuKCTT
uKLTKuKuKLTN
dnzKCTTuKuKTKuKN
654321
.ˆ)(ˆ)(
ˆˆˆ)(
)(ˆ)(
)ˆ(ˆˆˆ)ˆ(
..)(ˆˆˆˆ
(4.59)
where,
)ˆ(1 TF Th : is the approximation of heat flux normal to the boundary surface due to
conduction.
)ˆ(ˆˆˆ2 aavvTwvwavawh uKLTKuKuKLF : is the approximation of heat
flux normal to the boundary surface due to latent heat transfer by vapour movement.
zKCuKCTTF wwpwwwwpwrh ˆ)((3 : is the approximation of heat flux normal to
the boundary surface due to heat transfer by liquid movement.
TKuKuKCTTF vTwvwavawpvrhˆˆˆ)(4 : is the approximation of heat flux
normal to the boundary surface due to heat transfer by vapour movement resulting from
vapour pressure gradient.
aavpvrh uKCTTF ˆ)(5 : is the approximation of heat flux normal to the boundary
surface due to heat transfer by vapour movement resulting from bulk flow of air, and
aadapdarh uKCTTF ˆ)(6 : is the approximation of heat flux normal to the boundary
surface due to heat transfer by dry air movement resulting from an air pressure gradient.
By introducing the derivatives of the shape function equations from equation (4.5), (4.6)
and (4.7) into equation (4.58) and substituting fromequation (4.59), the final form of
equation (4.58) can be written as:
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
118
0
_
)(
654321e
e e e
eee
e
eee
ehhhhhh
T
aseTa
TseTT
TwseTw
T
ase
TaT
se
TTT
wse
TwT
eTwwPwr
aseT
TaseT
TTwseT
Tw
dFFFFFFNt
udNCNt
TdNCNt
udNCN
udNVNTdNVNudNVN
dzNKCTT
udNNKTdNNKudNNK
(4.60)
The spatial discretization of the governing differential equation for heat flow can be written
as:
TasTasTTwsTwas
Tas
TTws
Tw fuKTKuKt
uC
tT
Ct
uC (4.61)
where,
em
e eTw
TTw dNCNC
1 (4.62)
em
e eTT
TTT dNCNC
1 (4.63)
em
e eTa
TTa dNCNC
1 (4.64)
em
e eTw
TTw
TTw dNVNNKNK
1 (4.65)
em
e eTT
TTT
TTT dNVNNKNK
1 (4.66)
em
e eTa
TTa
TTa dNVNNKNK
1 (4.67)
m
e
m
e
ehhhhhh
TeTwwPwrT
eedFFFFFFNdzNKCTTf
1 1654321)(
(4.68)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
119
4.3.4 Spatial discretization of the solute transport governing equation
In this work the numerical solution for solute transport governing differential equation have
been stated for saturated and unsaturated soils as following:
4.3.4.1 Solute transport in saturated soils
The governing differential equation for solute transport in saturated soils was derived in
chapter 3 in equation (3.162) as:
0)()()( ccDvctc (4.69)
The primary unknowns can be approximated using the shape function approach for eight
node isoparametric element as:
ss cNcc8
1ˆˆ (4.70)
sscNcc8
1ˆ (4.71)
where, sc : is the nodal solute concentration.
By replacing the primary unknowns in equation (4.69) with their shape function
approximations from equations (4.70) and (4.71), equation (4.69) can be written as:
cRccDcvtc ˆ)ˆ()ˆ()ˆ( (4.72)
where,
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
120
cR : is the residual errors introduced by the approximation function.
Employing the Galerkin weighted residual approach to minimize the residual error
represented by equation (4.72) and integrating the equation over the spatial domain e
yields (AL-Najjar, 2006):
0ˆ)ˆ()ˆ()ˆ( e
er dccDcv
tcN (4.73)
In equation (4.73) it is convenient to integrate all the spatial derivatives by parts as
following:
The first term can be written as:
e
eer dNN
tcd
tcN
8
1
)ˆ( (4.74)
The advection which is represented by the second term can be written as:
e
ewy
e
ewx
er
ed
yNNcvd
xNNcvdNcv
8
1
8
1ˆ (4.75)
The dispersion-diffusion term which is represented by the third term can be written as:
e
ewwyx
e
ewwyy
e
ewwxy
e
ewwxx
er
e
dxN
yNcDd
yN
yNcD
dyN
xNcDd
xN
xNcDdNcD
8
1
8
1
8
1
8
1)ˆ(
(4.76)
Combining equations (4.75) and (4.76) yields:
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
121
0ˆˆ)( e
ewwyxwyywxywxxwywxr dcDDDDcvvN (4.77)
Discretization of equation (4.77) yields (Javadi et al, 2008):
0ˆ)(ˆˆ)(ˆ er
ewwxywxxwxewwxywxxwxr d
xNcDDcvcDDcvN (4.78)
0ˆ)(ˆˆ)(ˆ er
ewwyxwyywyewwyxwyywyr d
yNcDDcvcDDcvN (4.79)
Substituting equations (4.78) and (4.79) into (4.77), equation (4.77) can be rewritten as:
0ˆ)(ˆˆ)(ˆ
ˆ)(ˆˆ)(ˆ
er
ewwxywxxwx
er
ewwyxwyywy
ewwyxwyywyrewwxywxxwxr
dx
NcDDcvdy
NcDDcv
cDDcvNcDDcvN
(4.80)
The fourth term in equation (4.73) can be written as:
e
eer dNNcdcN
8
1ˆˆ (4.81)
Substituting the four terms from equations (4.74), (4.80) and (4.81) into (4.73), equation
(4.73) can be written as:
0ˆ
ˆ)(ˆˆ)(ˆ
ˆ)(ˆˆ)(ˆ
8
1
8
1
e
e
er
ewwxywxxwx
er
ewwyxwyywy
ewwyxwyywyre
wwxywxxwxre
e
dNNc
dx
NcDDcvdy
NcDDcv
cDDcvNcDDcvNdNNtc
(4.82)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
122
The integral in equation (4.82) can be evaluated for each element using the following
equation (AL-Najjar, 2006):
0)(8
1 eijijijer
eij cADcEvcB
yc
xcDvcNA
tc (4.83)
where, the elemental integrals in equation (4.83) are given by:
e
ij NNdA (4.84)
eij d
yNN
xNNB (4.85)
eij d
yN
xN
yN
xNE (4.86)
The discretized global finite element equation for a single component of solute transport
can be written as (Javadi et al, 2008):
cFHcdtdcM (4.87)
where,
ij
n
eA
tcM
1 (4.88)
ijij
n
eij cADcEvcBH
1 (4.89)
n
rc
eyc
xcDvcNF
1)( (4.90)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
123
4.3.4.2 Solute transport in unsaturated soils
The governing differential equation of solute transport in unsaturated soil was also defined
in chapter 3 in equation (3.175) as:
0)()()( ccDvctc (4.91)
By replacing the primary unknowns in equation (4.91) with their shape function
approximations from equations (4.70) and (4.71), equation (4.91) can be written as:
cRccDcvtc ˆ)ˆ()ˆ()ˆ( (4.92)
Employing the Galerkin weighted residual approach to minimize the residual error
represented by equation (4.92) and integrating the equation over the spatial domain e
yields (AL-Najjar, 2006):
0ˆ)ˆ()ˆ()ˆ( e
er dccDcv
tcN (4.93)
In equation (4.93) it is convenient to integrate all the spatial derivatives by parts as
following:
The first term can be written as:
e
eer dNN
tcd
tcN
8
1
)ˆ( (4.94)
The advection which is represented by the second term can be written as:
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
124
e
eaywy
e
eaxwx
er
ed
yNNcHvvd
xNNcHvvdNcv
8
1
8
1)()(ˆ (4.95)
The dispersion-diffusion term which is represented by the third term can be written as:
e
eaayxwwyx
e
eaayywwyy
e
eaaxywwxy
e
eaaxxwwxx
er
e
dxN
yNcHDD
dyN
yNcHDD
dyN
xNcHDD
dxN
xNcHDDdNcD
8
1
8
1
8
1
8
1
)(
)(
)(
)()ˆ(
(4.96)
By combining equations (4.95) and (4.96) as:
0ˆ
ˆ)()(
eayxayywxyaxxa
ewyxwyywxywxxwayaxwywxr
dcDDDDH
DDDDcvvHvvN (4.97)
Discretization of equation (4.97) yields (Javadi et al, 2008):
0ˆ)()(ˆ)(
ˆ)()(ˆ)(
er
eaaxyaxxwwxywxxaxwx
eaaxyaxxwwxywxxaxwxr
dx
NcHDDDDcHvv
cHDDDDcHvvN
(4.98)
0ˆˆ)(
ˆˆ)(
er
eaayxayywwyxwyyaywy
eaayxayywwyxwyyaywyr
dy
NcHDDDDcHvv
cHDDDDcHvvN
(4.99)
Substituting from equations (4.98) and (4.99) into equation (4.97), equation (4.97) can be
written as:
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
125
0ˆˆ)(
ˆ)()(ˆ)(
ˆˆ)(
ˆ)()(ˆ)(
er
eaayxayywwyxwyyaywy
er
eaaxyaxxwwxywxxaxwx
eaayxayywwyxwyyaywyr
eaaxyaxxwwxywxxaxwxr
dy
NcHDDDDcHvv
dx
NcHDDDDcHvv
cHDDDDcHvvN
cHDDDDcHvvN
(4.100)
The fourth term of equation (4.93) can be written as:
e
eer dNNcdcN
8
1ˆˆ (4.101)
Substituting the four terms from equations (4.94), (4.100) and (4.101) into (4.93), equation
(4.93) can be written as:
0ˆˆˆ)(
ˆ)()(ˆ)(
ˆˆ)(
ˆ)()(ˆ)(
8
1
8
1
eeraayxayywwyxwyyaywy
eraaxyaxxwwxywxxaxwx
aayxayywwyxwyyaywyr
aaxyaxxwwxywxxaxwxre
dNNcdy
NcHDDDDcHvv
dx
NcHDDDDcHvv
cHDDDDcHvvN
cHDDDDcHvvNdNNtc
ee
e
e
ee
(4.102)
The integral in equation (4.102) can be evaluated for each element using the following
equation (AL-Najjar, 2006):
0)(8
1 ee
eijijijrij cADcEvcB
yc
xcDvcNA
tc (4.103)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
126
where, the elemental integrals in equation (4.103) are given by:
e
ij NNdA (4.104)
eij d
yNN
xNNB (4.105)
eij d
yN
xN
yN
xNE (4.106)
The discretized global finite element equation for single component of solute transport
takes the form (Javadi et al, 2008):
cFHcdtdcM (4.107)
where,
ij
n
eA
tcM
1 (4.108)
ijij
n
eij cADcEvcBH
1 (4.109)
n
rc
eyc
xcDvcNF
1)( (4.110)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
127
4.3.5 Temporal discretization of the coupled fluid flow and heat transfer
The spatially discretized equations for coupled flow of water, air and heat are given by the
equations (4.20), (4.41) and (4.61) respectively. These equations can be combined in a
matrix form as:
T
a
w
as
ws
TTTaTw
aTaaaw
wTwaww
s
as
ws
TTTaTw
aTaaaw
wTwaww
fff
sT
uu
CCCCCCCCC
Tuu
KKKKKKKKK
... (4.111)
where,
tuu ws
ws (4.112)
tTT s
s (4.113)
tuu as
as (4.114)
Equation (4.111) can be rewritten in the following form (Istok, 1989):
FCK t)( (4.115)
where,
K : is the assemblage conductance matrix.
C : is the assemblage capacitance matrix.
F : is the vector of source or sink.
: is the vector of unknowns.
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
128
TTTaTw
aTaaaw
wTwaww
KKKKKKKKK
K ;
TTTaTw
aTaaaw
wTwaww
CCCCCCCCC
C ;
T
a
W
fff
F ;
s
as
ws
Tuu
Equation (4.115) is a system of ordinary differential equations, whose solution provides
values of unknowns at each node in the finite element mesh. This equation is highly non
linear and can only be solved using the iteration procedure. The finite difference approach
can be used to discretize the governing equations for fluid flow. Therefore, the time
descretisation of equation (4.115) can be written as (Istok, 1989):
tFFtKCtKC iiii 11 11 (4.116)
where,
i : is the iteration number.
: takes the value of 0, ½ or 1 for the forward, central, and backward difference schemes
respectively. A value of equal to 1 (for the backward difference scheme) has been used
in the current work as it proved its efficiency in solving highly non-linear equations.
Equation (4.116) can be rewritten as:
effieff FK 1 (4.117)
where
tKCKeff (4.118)
tFFtKCF iiieff 111 (4.119)
The iteration procedure commences with assuming an initial value for i as an initial
conditions. Therefore, the right hand side terms effF of equation (4.117) can be obtained at
the beginning of the time step, the first assumption for 1i is assumed to be the result of
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
129
the previous time step i , therefore the value of 1i for the first iteration can be obtained
by :
effeffi FK 11 (4.120)
The iterative solution proceeds till the problem converges. It is assumed that convergence
occurs when the following norm criterion is satisfied;
toleranceni
ni
ni
1
1'
11 (4.121)
where, n : is the time step level.
A solution of the fully coupled flow and heat transformation formulation can be achieved in
equation (4.116) which will give pore water pressure wu , pore air pressure au and
temperature T at various points within the soil and at different times, taking into account
the interaction between the flow of water, air and heat.
4.3.6 Temporal discretization of solute transport
The spatially discretized equation for solute transport in saturated and unsaturated soils is
given by equations (4.87) and (4.107) respectively. As explained earlier in equations
(4.115) and (4.116), the finite difference approach can be used to discretizee the governing
equations for solute transport. Therefore, equation (4.115) and (4.116) can be written for
solute transport as (Istok, 1989):
ct FMH ][ (4.122)
tFFtHMtHM ic
ic
ii 11 11 (4.123)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
130
where,
: is the vector of unknowns.
M , H and cF : are matrices defined by equations (4.88), (4.89) and (4.90) respectively.
Equation (4.123) can be rewritten as:
effc
ieff FH 1 (4.124)
where
tHMH eff (4.125)
tFFtHMF ic
ic
ieffc
111 (4.126)
Similarly, the iteration procedure commences with assuming an initial value for i as an
initial conditions, therefore, the right hand side terms effcF of equation (4.124) can be
obtained at the beginning of the time step. The first assumption for 1i is assumed to be
the result of the previous time step i , therefore the value of 1i for the first iteration can
be obtained by:
effc
effi FH 11 (4.127)
Similarly, the iterative solution proceeds till the problem converges. The convergence
occurs when the norm criterion is satisfied in equation (4.121).
A solution of the solute transport equation can be achieved in equation (4.123) which will
give the distribution of the solute concentrations c at various points within the unsaturated
soil and at different times, taking into account the interaction between the flow of water,
heat and various mechanisms of solute transport.
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
131
4.4 Coupling of fluid flow and solute transport equations
The numerical solution of coupled fluid flow and solute transport in soil is based on solving
the governing equations with appropriate boundary and initial conditions by an iterative
solution scheme as described in the following steps:
1- The required data including, control parameters, geometric data and initial and boundary
conditions for fluid flow, heat transfer and solute concentration are entered and checked.
2- The iterative routine starts at a given time step with an estimate of unknowns based on
the values of the previous time step.
3- Solution of the fluid flow and heat transfer equations provides the nodal values of pore
water pressure wu , pore air pressure au and temperature T at each point in the grid at
the next time step.
4- The nodal values of hydraulic head, fluid velocity, dispersion coefficient and fluid
density are computed according to the updated values of pressure. Through Darcy’s law
and a constitutive equation relating fluid density to salt concentration, the fluid flow and
solute transport equations are coupled and solved.
The hydraulic head wh is calculated as (Fredlund and Rahardjo, 1993):
zg
uhw
ww (4.128)
The fluid velocity is calculated according to Darcy’s law as (Darcy, 1856):
zuKvw
www (4.129)
The constitutive equation relating the fluid density to concentration can be used to
update the density as (Rastogi, 2004):
)1( Cf (4.130)
Chapter (4) Numerical solution of governing equations of coupled fluid flow and solute transport
132
where, f
fs and f , s are the freshwater and the seawater densities
respectively [M][L]-3.
5- Solution of solute transport equation provides the nodal values of concentration using
the updated parameters obtained in step 4.
6- The procedure (step 3 to step 5) is repeated by evaluating the concentration-dependant
parameters again with the updated values of concentration and solving the governing
equations at the same time level.
7- The cycle is repeated until the convergence criterion is satisfied and then the results are
recorded.
8- The results of the current time step are stored and used as initial values for the next time
step.
9- The cycle is repeated at each time step until the last time step.
The developed finite element model is an extension to a model that was developed by AL-
Najjar, (2006). The model has been extended to include density-dependent flow and an
algorithm has been implemented to incorporate density-dependency of flow in the original
model. The solute transport formulation has been extended to include new transport
mechanisms. The model has been applied to study seawater intrusion and the effect of
climate change and sea level rise on coastal aquifers. Also, a simulation-optimisation model
has been developed by integrating the FE model and a genetic algorithm to optimize
arrangements for SWI control. A flow chart of the numerical solution algorithm is shown
in Figure 6.1.
The application of the proposed model requires sufficient and accurate data. The types of
data required for the application of the proposed model are mainly based on the
representation of the fluid flow and solute transport process. Thus, the data file used by
both coupled flow and transport analyses contains the initial values of pore water pressure
wou , pore air pressure aou , temperature oT and solute concentration oc at all nodes of the
domain, longitudinal and transverse dispersivity values, molecular diffusion coefficient,
density, porosity of the soil and the boundary conditions on boundaries of the domain for a
particular site.
Chapter (5) Validation and application of the simulation model
133
Validation and application of the simulation model
5.1 Introduction
A finite element model for saturated/unsaturated fluid flow and solute transport (SUFT) has
been developed. The developed model is transient, density-dependant and the dispersion is
velocity dependant. The coupled fluid flow and solute transport model can handle a wide
range of real-world problems including the simulation of groundwater flow and solute
transport separately and coupled fluid flow and solute transport in addition to saltwater
intrusion in coastal aquifers. This chapter presents the validation and application of the
developed model. The model is validated against a number of problems from literature to
examine its capabilities to handle solute transport problems through saturated/unsaturated
porous media with density-dependant flow. Then, the model is applied to a number of real
case studies for predicting saltwater intrusion in coastal aquifers in different locations of the
world.
5.2 Calibration, validation, and verification of the model
Numerical models may be used to predict fluid flow or solute transport in soils. These
models may also be used to evaluate different remediation alternatives. However, errors
and uncertainties in a fluid flow and solute transport analysis make any model prediction no
better than an approximation. Therefore, all model predictions should be expressed as a
range of possible outcomes that reflect the assumptions involved and uncertainty in model
input data and parameter values. Numerical model application can be considered to be two
distinct processes. The first process is model development resulting in a software product,
and the second process is application of that product for a specific purpose. The developed
model should be calibrated and validated to ensure its accuracy before the application.
Chapter (5) Validation and application of the simulation model
134
Calibration of a numerical model refers to the process in which model input parameters are
adjusted, either manually or through formal mathematical procedures, until the model
output matches the field-observed conditions satisfactorily. The model output variables can
be hydraulic heads, flow rates, solute concentrations, or mass removal rates, depending on
the objectives of the simulation. In practical model applications, model input parameters
are never completely defined and are always associated with various uncertainties, no
matter how many measurements have been made or how thoroughly site conditions have
been characterised (Gunduz, 2004). Verification of a numerical model has been defined as
the process in which the calibrated model is shown to be capable of reproducing a set of
field observations independent of those used in model calibration. Validation of a
numerical model is the process in which the calibrated model is shown, through a postaudit,
to be capable of predicting future conditions with sufficient precision (Anderson and
Woessner, 1992). Once the developed model is calibrated, verificated and validated, it can
be used for predictive simulations.
The fluid flow approaches to the governing equations suggest a natural way to define
dimensionless groups that characterize the advection and diffusion. When numerically
solving the solute transport equation, the appropriate dimensionless groups are Courant and
Peclet numbers. These dimensionless groups have been found to be useful aids in
characterizing the behavior of the numerical methods and for choosing appropriate
discretizations. In order that accurate solutions are obtained, several guides on the
discretization should be followed. Daus et al. (1985) used numerical experiments to
demonstrate the behaviour of the finite element method in cases where the Peclet and
Courant number criteria are not met. They showed that if the Peclet and Courant number
criteria are not met there are no general rules for predicting the solution behaviour. Courant
and Peclet numbers are described below:
Chapter (5) Validation and application of the simulation model
135
Courant Number
Courant number is described mathematically as a function of the velocity of the fluid v ,
time step and spatial grid sizes x as:
xtvCr (5.1)
Courant number should be chosen )1( rC so that the equation satisfies the the Courant-
Friedrichs-Lewy condition (CFL condition) (Courant et al., 1928). In mathematics, the
Courant-Friedrichs-Lewy condition is a condition for certain algorithms for solving partial
differential equations to be convergent. As a consequence, the time step must be less than a
certain time in many explicit time-marching computer simulations, otherwise the
simulation will produce wildly incorrect results.
Peclet Number
Peclet number is a relationship between the advective and diffusive components of solute
transport and is expressed as the ratio of the product of the average interstitial velocity v ,
times the characteristic length, divided by the coefficient of molecular diffusion mD :
me D
xvP (5.2)
Small values indicate diffusion dominance whereas large values indicate advection. For
purely advective problems Peclet number will equal to infinity as:
Le
xD
xvP (5.3)
where, L is the longitudinal dispersivity (m). Peclet number should be chosen )2( eP .
The error measure shows that while the oscillations are smaller for larger Peclet numbers,
the solution is not much more accurate (Binning, 1994).
Chapter (5) Validation and application of the simulation model
136
5.3 Numerical results
Coupled fluid flow and solute transport models play an important role in many areas of
engineering. The analysis of saltwater intrusion into coastal aquifers and salinization of
groundwater is one of these areas. Therefore, a clear understanding of solute transport and
the relevant transport processes is important for engineers. The numerical modeling can be
used to further understanding of the seawater intrusion mechanism. Furthermore, numerical
predictions could be used as an alternative means to assist coastal water resources planning
in lieu of field data. In the following sections the developed finite element model (SUFT) is
validated by application to a number of examples from the literature, including the
simulation of groundwater flow alone, solute transport alone and coupled fluid flow and
solute transport, and saltwater intrusion in coastal aquifers.
5.3.1 Example (1): Transient groundwater flow
This example presents a transient flow in a confined aquifer in one dimension. A transient
flow problem is one in which the unknown variable is time dependant. Transient problems
are also called time dependant, nonequilibrium, or unsteady problems. This example
represents a transient groundwater flow in a confined aquifer shown in Figure 5.1. The
aquifer length is 100 m and the depth is 10 m. The domain has been divided into ten
elements (10x10) m. The purpose is to simulate changes in head with time due to drop the
water level in the reservoir from 16 m to 11 m. The boundary conditions are no flow along
the top and bottom boundaries, 16 m head on the left boundary and 11 m head on the right
boundary. The initial conditions are 16 m head everywhere in the aquifer. The boundary
conditions are shown in Figure 5.1 (Wang and Anderson, 1982).
Chapter (5) Validation and application of the simulation model
137
Figure 5.1 Boundary and initial conditions of example (1)
The developed model (SUFT) has been used to calculate the distribution of head in the
aquifer at each time step for total time of 500 min and t equal to 5 min. The results are
shown in Figure 5.2.
Figure 5.2 Head distribution at different times in the reservoir
h1 = 16
h2 = 11 m
x = 0 x = 100 m
N o flow
N o flow y = 10
y = 0
X
Y
Chapter (5) Validation and application of the simulation model
138
The results are compared with those reported by Wang and Anderson (1982), who
presented a solution to this problem using the finite difference and finite element methods
using rectangular elements. Figure 5.3 shows the results of the current model (SUFT)
compared with the results of Wang and Anderson (1982). As shown the current results
match with, a high accuracy, the published results for this problem.
Figure 5.3 Comparison between the current model and Wang and Anderson results for
head distribution in the reservoir
5.3.2 Example (2): Transient solute transport
The current finite element model is applied to simulate transient solute transport in
saturated soil in the confined aquifer presented in example (1). The boundary conditions are
chosen to be c=10 on the left boundary and c=0 on the right boundary as shown in Figure
5.4. The initial conditions are c=0 everywhere in the aquifer. The flow velocity is equal to
0.1m/day, the longitudinal and the transverse dispersion coefficients are 1 and 0.1 m
respectively (Wang and Anderson, 1982). The purpose of this example is to simulate
changes in solute concentration c through time in the aquifer.
Chapter (5) Validation and application of the simulation model
139
Figure 5.4 Boundary and initial conditions of example (2)
The developed model has been used to calculate the distribution of solute concentration in
the aquifer at each time step for total time of 500 min and t equal to 5 min. The results of
the current code are shown in Figure 5.5.
Figure 5.5 Solute concentrations distribution at different times in the reservoir
C= 10 C = 0
x = 0 x = 100 m
y = 10
y = 0
Y
X
Chapter (5) Validation and application of the simulation model
140
Wang and Anderson (1982) presented a finite element model to solve advective-dispersive
transport. A comparison of the current results with Wang and Anderson (1982) at time
t=10, 50, 100, 200, 300, and 500 sec, is shown in Figure 5.6. The results show a good
agreement with the other results and the difference is due to using different techniques and
mesh size.
Figure 5.6 Comparison between the current model and Wang and Anderson results for
solute concentrations distribution in the reservoir
Chapter (5) Validation and application of the simulation model
141
5.3.3 Example (3): Saltwater intrusion in confined aquifer (Henry’s
problem)
This example concerns couple groundwater flow and solute transport in a confined aquifer.
In this example the developed model is applied to one of the most popular benchmark
problems in seawater intrusion into a confined coastal aquifer, widely known as Henry’s
saltwater intrusion problem (Henry, 1964). This case involves seawater intrusion into a
confined aquifer studied in vertical cross section perpendicular to the coast. Henry
developed the first analytical solution including the effect of dispersion in confined coastal
aquifer under steady-state conditions. He assumed a constant dispersive coefficient in the
aquifer.
Henry’s problem was studies by Pinder and Cooper (1970), Lee and Cheng (1974), Segol et
al. (1975), Huyakorn et al. (1987), Frind (1982), Cheng et al. (1998) and Rastogi et al.
(2004). The aquifer under consideration is uniform and isotropic and bounded above and
below by impermeable strata. The aquifer is exposed to stationary seawater intrusion body
on the right side and a constant freshwater flux on the left side. The aquifer boundary
conditions allow convective mass transport out of the system over the top portion (20%).
Figure 5.7 shows the aquifer and the boundary conditions.
Figure 5.7 Boundary conditions of Henry’s problem
1.0
2.0
Impermeable
(C=1)
(C=0)
QIN
Impermeable
20%
80%
Chapter (5) Validation and application of the simulation model
142
The objective of this example is to compare the results of the current model with the
previous published results to test the capability of the current model to give acceptable
results for the solution of Henry’s problem (1964).
Boundary Conditions
The boundary conditions of Henry’s problem are defined in Figure 5.7. No flow occurs
across the top and bottom boundaries. The freshwater source is set along the left vertical
boundary and is implemented by employing source nodes along the left vertical boundary,
which inject freshwater at a rate of inQ , and concentration of inc . Specified pressure is set at
hydrostatic seawater pressure with ( s =1025 kg/m3) along the right vertical boundary
through the use of specified pressure nodes. The soil parameters used in Henry’s problem
are shown in Table 5.1.
Table 5.1 The parameters used in Henry’s problem
mD : coefficient of water molecular diffusion (m2/sec) 6.6x10-6
T : transverse dispersivity (m) 0.0
L : longitudinal dispersivity (m) 0.0
n : porosity 0.35
: fluid viscosity 0.001
inQ : inland fresh water flux (m3/sec) 6.6x10-5
g : gravitational acceleration (m/sec) 9.81
w : density of fresh water (kg.m-3) 1000
s : density of sea water (kg.m-3) 1025
K : hydraulic conductivity (m/sec) 1x10-2
k : permeability (m2) 1x10-9
Chapter (5) Validation and application of the simulation model
143
Initial Conditions
Freshwater concentration and natural steady-state pressures are initially set everywhere in
the aquifer. The natural initial pressure values are obtained through an extra initial
simulation that calculates steady pressures for the conditions of freshwater concentration
(c=0) throughout, with the boundary conditions described above.
Simulation setup
Dimensions of the problem are selected to allow for simple comparison with the steady-
state dimensionless solution of Henry (1964), and with a number of other published
simulation models. The aquifer region is represented by quadrilateral isoparametric (eight-
nodded) elements, the mesh consisting of 661 nodes and 200 elements, each of size 0.1 m
by 0.1 m. Mesh thickness is 1.0 m and mesh width is 2.0 m. A total simulation time of
t=100.0 min, is selected and the time step size is t = 60 sec. To test and compare the
model under consideration, the results are compared with four cases from the literature as
following:
1- Steady-state constant dispersion,
2- Steady-state variable dispersion,
3- Transient constant dispersion, and
4- Transient variable dispersion.
The results of the current model together with results of other models are presented in the
following sections. Seawater wedge is chosen to be represented by ( 5.0/ *cc ) isochlor,
which is an approach adapted by many researchers.
5.3.3.1 Steady-state constant dispersion
In the case of constant dispersion, Henry’s solution assumes the dispersion is presented by
a constant coefficient of diffusion, rather than by velocity-dependant dispersivity. In the
literature, two values for molecular diffusion coefficient (Dm) have been used; 18.8571x10-6
and 6.6x10-6 [m2/sec]. The total dispersion coefficient D, is equal to the product of porosity
and molecular diffusivity, D = nDm (Voss and Provost (2003). To match the values of
Henry’s solution, values of D = 6.6x10-6 and 2.31x10-6 , are used. The results of the current
code compared with INTERA and SUTRA codes (Voss and Provost, 2003) for the higher
Chapter (5) Validation and application of the simulation model
144
value of diffusion Dm = 18.8571x10-6 and D = 6.6x10-6 with both the longitudinal and
transverse dispersivities set to zero are shown in Figure 5.8.
Figure 5.8 0.5 Isochlor for steady-state constant dispersion (D = 6.6x10-6)
For the lower value of diffusion Dm = 6.6x10-6 and D = 2.31x10-6 , the results are shown in
Figure 5.9. The results are compared with Pinder and Cooper (1970) (method of
characteristics), Segol et al. (1975) (finite element method), Frind (1982) (finite element
method), and SUTRA (finite element-finite difference method), (Voss and Provost (2003).
Figure 5.9 0.5 Isochlor for steady-state constant dispersion (D = 2.31x10-6)
Chapter (5) Validation and application of the simulation model
145
From Figures 5.8 and 5.9, it can be seen that the results obtained using the current code
match well with the results from INTERA, SUTRA and other codes.
5.3.3.2 Steady-state variable dispersion
In Henry’s solution and those of some other researches it was assumed that dispersion is
represented by a constant coefficient of diffusion, rather than being velocity-dependent.
However, Rastogi et al. (2004) considered the dispersion coefficients to be velocity-
dependent under steady state conditions and selected values for longitudinal and transverse
dispersivities as 0.5 and 0.1 m respectively. The results of the current model are compared
with some known solutions. Figure 5.10, shows the results of position of the 0.5 isochlor
obtained by the present code compared with the results of Henry (1964), Pinder and Cooper
(1970), Lee and Cheng (1974) and Rastogi et al. (2004). The present results match nicely
with the existing solutions. The shape of the interface is S-type and similar to the known
shapes (Rastogi et al., 2004).
Figure 5.10 0.5 Isochlor for steady-state variable dispersion
Chapter (5) Validation and application of the simulation model
146
5.3.3.3 Transient constant dispersion
In this case, the developed model results are compared with a two-dimensional finite
volume unstructured mesh method (FVUM), which was developed by Liu et al. (2001) for
analyzing the evolution of the saltwater intrusion into coastal aquifer systems. The model of
Liu et al. (2001) considerd transient groundwater flow and solute transport in a coastal
aquifer and was validated by comparing the results for Henry’s problem with those of
Huyakorn et al. (1987) and Cheng et al. (1998). Dimensions of the problem are selected to
allow comparison with the solution of Liu et al. (2001). The thickness is 100.0 m and width
is 200.0 m. The domain consists of twenty by ten elements, each of size 10.0 m by 10.0 m,
(NN=661, NE=200). A total simulation time of t=6000 days, is used with time step size of
t = 5 days (Liu et al., 2001).
Two cases are selected for the validation; constant dispersion and variable dispersion. In the
first case, a constant dispersion value is used based on the molecular diffusion coefficient
set equal to 6.6 x10-2 (m2/d), and both the longitudinal and transverse dispersivities were set
to zero. Figure 5.11, shows the results in terms of position of 0.5 isochlor obtained by the
present model compared with the results of Huyakorn et al. (1987), Cheng et al. (1998) and
Liu et al. (2001).
Figure 5.11 0.5 Isochlor for transient constant dispersion
Chapter (5) Validation and application of the simulation model
147
5.3.3.4 Transient variable dispersion
For the variable dispersion case, the dispersion coefficient is considered velocity-dependant
rather than constant. The diffusion coefficient is set to zero and both the longitudinal and
transverse dispersivities are 3.5 m. Figure 5.12 shows the results of the current code
compared with Liu et al. (2001). It can be observed from Figures 5.11 and 5.12 that the
results are in good agreement with previously published solutions by Liu et al. (2001).
Figure 5.12 0.5 Isochlor for transient variable dispersion
5.3.3.5 Effect of variability in dispersion and density on SWI
To illustrate the effect of dispersion coefficient on the distribution of solute concentration
and saltwater intrusion in coastal aquifers, the results of the finite element analyses were
obtained for two cases of constant dispersion coefficient and variable dispersion coefficient
and compared with those of Huyakorn et al. (1987), Cheng et al. (1998) and Liu et al.
(2001) as shown above. In the case of variable dispersion, the dispersion coefficient is
considered to be velocity-dependant rather than constant. Figure 5.13 shows a comparison
between 0.5 isochlors due to constant dispersion and variable (velocity-dependant)
dispersion coefficient. As shown in Figure 5.13 the intrusion of saltwater increases in land
in the case of variable dispersion coefficient (velocity-dependant) (Liu et al., 2001).
Chapter (5) Validation and application of the simulation model
148
Figure 5.13 0.5 Isochlor for constant and variable dispersion coefficient
To illustrate the effect of fluid density on the distribution of solute concentration and
saltwater intrusion in coastal aquifers, the results of the finite element analyses were
obtained for two cases, constant density and variable density. Figure 5.14 shows a
comparison between 0.5 isochlor due to constant density and variable density. The densities
of fresh water f and seawater s are considered 1.0 and 1.025 Mg/m3 respectively. Due
to the difference between the two densities being very small as 0.025, the effect of variation
in density is small. Figure 5.14 show that the increase in sweater density decreases the
intrusion of saltwater.
Figure 5.14 0.5 Isochlor for constant and variable density
Chapter (5) Validation and application of the simulation model
149
5.4 Case studies
In the previous section, the developed model was verificated against a number of examples
to show its ability to simulate groundwater flow and solute transport separately and coupled
fluid flow and solute transport in soil. This section presents the application of the model to
four case studies to predict seawater intrusion into coastal aquifers in different locations of
the world. The first case is a hypothetical case from the literature that was investigated by
Kawatani (1980). The second is saltwater intrusion into Madras aquifer in India (Sherif et
al., 1990). The third is saltwater intrusion in Biscayne aquifer at Coulter area near Miami,
Florida, USA (Lee and Cheng, 1974) and finally saltwater intrusion in Gaza aquifer in
Palestine (Qahman and Larabi, 2006).
5.4.1 Case (1): A hypothetical case study
In this case the developed model is applied to a hypothetical case from the literature. The
problem was investigated by Kawatani (1980). He considered the case of an unconfined
coastal aquifer with a channel carrying fresh water on the land side (Figure 5.15). The
water infiltration from the channel creates a constant water table above the sea level which
can help in controlling saltwater intrusion. Kawatani (1980) presented a finite element
model to study saltwater intrusion and Sherif et al. (1990) presented a steady-state solution
to this problem.
Site description
The region of the study is 500 x 20 m. The water table on the land side (fresh water level in
the channel) is 0.35 cm above the sea level leading a hydraulic gradient of 7 x 10-4. The
hydraulic conductivities in x-direction xK and y-direction yK are 0.06 cm/min and 0.006
cm/min respectively. The effective porosity is 0.25 and dispersion coefficients in x-
direction xD and y-direction yD are 5 and 0.5 cm2/min respectively. The longitudinal and
transverse dispersivities L and T are 5 and 2.5 m. The density of fresh water f and
seawater s are 1.0 and 1.025 t/m3 respectively (Sherif et al., 1990).
Chapter (5) Validation and application of the simulation model
150
Figure 5.15 Problem description of unconfined coastal aquifer with fresh water channel at
land side
Boundary conditions
In this study, the same domain with the same hydraulic parameters and boundary conditions
is considered to compare the results with those of Kawatani (1980) and Sherif et al. (1990).
The aquifer under consideration is homogenous and isotropic and bounded above and
below by impermeable strata. The aquifer is exposed to stationary seawater intrusion body
on the right side and a constant freshwater head from the channel on the left side. The
aquifer boundary conditions allow convective mass transport out of the system over the top
portion (20%). Figure 5.16 shows the boundary conditions applied in this case. The domain
was represented by quadrilateral isoparametric (eight-noded) elements, the mesh consisting
of 400 elements and 1409 nodes. The time step of 5 days is used with total time of 250
days.
Sea
Channel
Water table
Chapter (5) Validation and application of the simulation model
151
Figure 5.16 Boundary conditions of the hypothetical case
Results
Figure 5.17 shows 0.5, 0.25, 0.1, 0.05 and 0.005 isochlor contours obtained by the current
model and those of Kawatani (1980) and Sherif et al. (1990). The agreement between the
results is good. A possible explanation for the discrepancies between the results is the
different numerical schemes and different grid systems used.
Figure 5.17 0.5, 0.25, 0.1, 0.05 and 0.005 Isochlor contours of the hypothetical case
20
500 Impermeable
(C=1) (C=0)
hp f
20%
80%
Impermeable
hp s
Sea side Land side
Chapter (5) Validation and application of the simulation model
152
5.4.2 Case (2): Madras aquifer, India
This case presents a real problem of saltwater intrusion in Madras coastal aquifer in south
India. Saltwater intrusion in Madras coastal aquifer has been studied by a number of
researches such as; Rouve and Stroessinger (1980) and Sherif et al. (1990) who applied
finite element models to predict the saltwater intrusion in this aquifer. The aquifer thickness
at the sea boundary is 30 m. Because the thickness is small, the limit up to which the
influence of hydrodynamic dispersion can be expected is about 3.0 km (Sherif et al. (1990).
Site description
The study region which was considered by Rouve and Stroessinger (1980) and Sherif et al.
(1990) is 2600 x 30 m. a coefficient of permeability of 3x10-3 m/sec and porosity of 0.35
were considered. The longitudinal and transverse dispersivities L and T were taken as
66.6 and 6.6 m, respectively. The density of fresh water f and seawater s are 1.0 and
1.025 t/m3. Molecular diffusion was set equal to 1x10-6 m2/sec.
Boundary conditions
The same domain with the same hydraulic parameters and boundary conditions are
considered to compare the results with Rouve and Stroessinger (1980) and Sherif et al.
(1990). The aquifer is leaky and heterogeneous but for simplicity Rouve and Stroessinger
assumed that the aquifer is confined and homogeneous. The piezometric heads at the
seaside was set equal to 40 m and 41 m at the land side. Figure 5.18 shows the boundary
conditions applied in this case. The domain was represented by quadrilateral isoparametric
(eight-noded) element, the mesh consisting of 390 elements and 1437 nodes.
Chapter (5) Validation and application of the simulation model
153
Figure 5.18 Boundary conditions of Madras aquifer
Results
The developed model is applied to the idealized representation of the Madras aquifer. No
observations about salinity distribution in the Madras aquifer were available. The results
are compared with those of Rouve and Stroessinger (1980) and Sherif et al. (1990). Figure
5.19 shows 35, 17.5, 10, 5 and 2.0 equi-concentration lines obtained by the current model
and those of Rouve and Stroessinger (1980) and Sherif et al. (1990). The results of the
current code match well with the other results. Rouve and Stroessinger (1980) and Sherif et
al. (1990) used steady-state finite element analysis for the current case and considered
velocity-independent dispersion whereas in the current study transient variable-density flow
and velocity-dependant dispersion coefficients are considered which resulted in further salt
advancement inland as shown in Figure 5.19.
30
2600
Impermeable
(C=1) (C=0)
hp f hp s
Sea side Land side
(41.00) (40.00)
piezometric head
Chapter (5) Validation and application of the simulation model
154
Figure 5.19 35, 17.5, 10, 5 and 2.0 equi-concentration lines in Madras aquifer
5.4.3 Case (3): Biscayne aquifer, Florida, USA
Lee and Cheng (1974) presented steady-state 2-D finite element model using triangular
elements to study saltwater encroachment to Biscayne aquifer at Coulter area, Florida,
USA, where a field investigation had been made by Kohout (1960a, b). The developed
model is applied to simulate saltwater intrusion into Biscayne aquifer and the results are
compared with those reported by Lee and Cheng (1974) and the observations of Kohout
(1960a, b).
Site description
The geologic and hydraulic aspects of Biscayne aquifer have been investigated and reported
by Kohout (1960a, b). The aquifer average depth is 100 ft below mean sea level. It is
observed that the transition zone is quite thick and the toe of the saltwater is approximately
1600 ft from the shore, whereas the 0.5 isochlor (9500 ppm chloride) is located at the
bottom of the aquifer about 1400 ft from the bay. The study region which is considered in
this study is 300 x 30 m (Lee and Cheng (1974) and Kohout (1960a, b)). A coefficient of
permeability of 3x10-10 m/sec and porosity of 0.39 are considered. The longitudinal and
Chapter (5) Validation and application of the simulation model
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transverse dispersivities L and T are taken as 5.0 and 0.5 m, respectively. The density
of fresh water f and seawater s are 1.0 and 1.025 t/m3. The dispersion coefficients in x-
direction xD and y-direction yD are velocity-dependant.
Boundary conditions
The same hydraulic parameters and boundary conditions were considered to compare the
results with Lee and Cheng (1974) and Kohout (1960a, b). The schematic sketch of the
Biscayne aquifer at Coulter area and the corresponding boundary conditions applied in this
case are shown in Figure 5.20. The domain is represented by quadrilateral isoparametric
(eight-noded) element, the mesh consisting of 250 elements and 861 nodes.
Figure 5.20 Schematic sketch and boundary conditions of Biscayne aquifer, Florida
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(c=1) (c=0)
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Biscayne Bay Water table
Land surface
Base of Biscayne aquifer
Chapter (5) Validation and application of the simulation model
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Results
The flow in Biscayne aquifer is studied to test the applicability of the present model to
simulate saltwater intrusion into coastal aquifers. The results were compared with those of
Lee and Cheng (1974) and the observations of Kohout (1960a, b). Figure 5.21 shows 1,
0.75, 0.5 and 0.25 isochlor contours obtained by the current model and those of Lee and
Cheng (1974), whose compared their results with the observations of Kohout (1960a, b).
Good qualitative agreement exists between the current predictions and Lee and Cheng and
Kohout’s observations.
Figure 5.21 1, 0.75, 0.5 and 0.25 isochlor counters in Biscayne aquifer, Florida
Chapter (5) Validation and application of the simulation model
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5.4.4 Case (4): Gaza Aquifer, Palestine
A real world problem is addressed in this case, the coastal aquifer of Gaza Strip which is
intruded by saltwater. Gaza strip has a coastline of about 45 km and a total area of about
365 km2. Gaza strip is home to about 1.5 million Palestinians. The aquifer is the only
source of fresh water in Gaza, and provides more than 95% of the water consumed in the
Strip for agriculture, domestic and industrial uses. Water is obtained by pumping of more
than 3000 wells, with a total estimated annual yield of about 140 million cubic meters
(Qahman and Larabi, 2006). Gaza aquifer is a part of the shallow sandy coastal aquifer
which stretches from Israel through Gaza into the Sinai Peninsula in Egypt as shown in
Figure 5.22.
Figure 5.22 Location map of Gaza strip
(Source: EPD/IWACO-EUROCONSULT 1996)
Section I Jabalya
Section II Wadi Gaza
Section III Khan Younis
Chapter (5) Validation and application of the simulation model
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A very critical and rapidly worsening component of humanitarian crisis in the Gaza strip is
the water crisis. Over-abstraction of water from the aquifer over period of time (several
decades) has caused severe depletion and saltwater intrusion from the Mediterranean Sea.
The flow of water in the aquifer is naturally toward the sea, though in many places the flow
has been reversed due to over-abstraction of groundwater which has led to the intrusion of
seawater into the aquifer. Another major problem for the Gaza aquifer is pollution;
contamination consists of fertilizers and pesticides and untreated sewage. The pollution,
exacerbated by the conflict's impact on Gaza's waste-treatment facilities, has resulted in
widespread contamination of the aquifer. Moreover, climate change predictions suggest an
imminent catastrophe; as temperature rises, the droughts would exacerbate, and as the sea
level rises, saltwater intrusion would speed up. Rising sea level poses another challenge to
the Gaza population as it might lead to the submerging of much of Gaza's territory.
Saltwater intrusion and pollution of the aquifer affect adversely both the human population
of the Gaza Strip and its environment. High salinity has been directly linked to a
skyrocketing rate of kidney diseases in Gaza. Saltwater intrusion poses a great threat to the
municipal supply and industrial growth in the Strip (Qahman, 2004).
Seawater intrusion in Gaza aquifer has been studied by a number of researchers (e.g.,
Yakirevich et al. (1998); Qahman and Zhau (2001); Moe et al. (2001); Qahman and Larabi
(2003a, b) and Qahman and Larabi (2006)). Yakirevich et al. (1998) used SUTRA code and
Qahman and Larabi (2003b) used SEAWAT code to simulate saltwater intrusion in the
Gaza aquifer. Numerical simulations showed that the rate of seawater intrusion during
1997-2003 was expected to be 20-45 m/year. The developed model considering coupled
flow and solute transport with variable density effects on groundwater flow, is applied to
simulate groundwater flow and saltwater intrusion in Gaza aquifer. Coupling flow and
transport computations allows the effects of fluid density gradients associated with solute
concentration gradients to be incorporated into groundwater flow simulations.
Chapter (5) Validation and application of the simulation model
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Description of the study area Gaza Strip is a part of the Mediterranean coastal plain between Egypt and Israel. It forms a
long and narrow rectangle shape of 45 km long and width of 5 to 10 km. Gaza Strip is a
foreshore plain gradually sloping westward. The groundwater flow is mainly westward
toward the Mediterranean Sea and the maximum thickness of the saturated zone is 120 m
near the sea. Three vertical cross sections have been selected to study the intrusion of
seawater in the aquifer at Jabalya, Wadi Gaza and Khan Yunis (Figure 5.22). The first
section at Jabalya is located in the north of Gaza Strip with a length of 5000 m and average
depth of 100 m. The second section at Wadi Gaza is located in the middle of Gaza Strip
with a length of 2000 m and average depth of 100 m. The third section at Khan Yunis is
located in the south of Gaza Strip with a length of 5000 m and average depth of 100 m with
inclined top and bottom. The aquifer is considered confined, homogeneous and isotropic
with respect to freshwater hydraulic conductivities, molecular diffusion, and longitudinal
and transverse dispersivities. The aquifer system is subjected to seawater intrusion along
the sea face boundary and a uniformly distributed lateral flow along the inland face, a
constraint head of 1m is considered. The hydraulic conductivities xK and yK are 0.2
m/day and 0.1 m/day respectively. The effective porosity is 0.35 and the longitudinal and
transverse dispersivities L and T are 50 and 0.1 m respectively. The density of fresh
water f and seawater s are 1.0 and 1.025 t/m3 respectively (Qahman, 2004).
Boundary conditions
The flow boundary conditions consist of impermeable boundaries along the top and the
bottom of the aquifer. Constant head and concentration are specified to the model along the
coast. Hydrostatic pressure is assumed along the vertical boundary of the sea side. The
aquifer is charged with freshwater at constant flux from the inland side. The prescribed
constant concentration (TDS) of 35 kg/m3 is considered (Qahman, 2004).
Chapter (5) Validation and application of the simulation model
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Results
The developed model is applied to simulate saltwater intrusion in the selected three vertical
cross sections in order to predict the intrusion of seawater in different locations in the Gaza
aquifer. The concentration distributions for the three cross sections are presented in Figures
5.23, 5.24 and 5.25.
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Chapter (5) Validation and application of the simulation model
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The results of the present model show that the aquifers is subject to a sever seawater
intrusion problem and groundwater is unsuitable for human uses. At Khan Yunis, isoline 2
(2000 ppm) intruded inland to a distance of about 2.0 km measured along the bottom
boundary (Figure 5.25). On the other hand, at Jabalia, the same isoline intruded inland to a
distance of about 3.0 km (Figure 5.23). The intrusion reduction at Khan Yunis might be
attributed to the upward slope toward inland side at this section. At Wadi Gaza, isoline 2
(2000 ppm) intruded inland to a distance of about 1.2 km, measured along the bottom
boundary (Figure 5.24).
The results of the geophysical survey obtained by PWA/CAMP (2000), indicated that the
extent of saltwater intrusion ranged between 1 km and 2 km at Khan Yunis cross section.
Qahman and Larabi (2006) used SEWAT code to simulate saltwater intrusion in Gaza
aquifer. The estimated extent of seawater wedge from seawater intrusion until year 2003
along Khan Yunis ranged between 1 and 2 km which is similar to the results obtained by
the geophysical survey (see Figure 5.26). Also the results of the present model indicated
that the intrusion of seawater at this section is about 2 km (Figure 5.25).
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Chapter (5) Validation and application of the simulation model
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Figure 5.26 Isolines of TDS concentration (kg/m3) calculated by SEAWAT along
Khan Yunis (Qahman and Larabi, 2006)
Qahman and Larabi (2006) also used SEWAT to simulate saltwater intrusion along Jabalia
and the estimated extent of seawater wedge was about 3 km until year 2003 (Figure 5.27)
which is similar to the results obtained by present model (Figure 5.23).
Figure 5.27 Isolines of TDS concentration (kg/m3) calculated by SEAWAT along
Jabalya (Qahman and Larabi, 2006)
Chapter (5) Validation and application of the simulation model
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The results of the developed model gave a good agreement with the results of previous
models applied to Gaza aquifer (e.g. Qahman and Zhou (2001); Qahman and Larabi
(2003a, b); Qahman and Larabi (2006) and results of the geophysical survey obtained by
PWA/CAMP (2000)). The discrepancy in results of the current model and the work of
Qahman and Larabi (2006) is due to the fact that they have considered the aquifer to be
divided to three sub-aquifers overlaying each other and separated by impervious or semi-
previous clayey layers. For simplicity the aquifer has been considered homogenous in the
current study. However, the results generally match those of Qahman and Larabi, with the
difference being in the shape of the contour lines. Qahman and Larabi (2006) also used
SEWAT code to simulate saltwater intrusion in the Gaza aquifer in horizontal plane using
two different scenarios of pumping. The special distribution of TDS concentration
predicted by SEWAT is shown in Figure 5.28. It is shown that the intrusion of seawater at
Khan Younis is about 2 km, Jabalya about 3 km and Wadi Gaza about 1.2 km which is
consistent with the results of the present model. The results of the present model in vertical
view have been compared with SEWAT results in horizontal view. The comparison
between the two models shows a good agreement in terms of concentration distribution in
vertical sections obtained by the current model and horizontal section obtained by SEWAT.
Figure 5.28 Extant of TDS concentration calculated by SEAWAT
(Qahman and Larabi, 2006)
Chapter (5) Validation and application of the simulation model
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Seawater intrusion in the Gaza aquifer was simulated using the developed model in vertical
sections and concentration contours of salt distribution in the aquifer were presented and
compared with the results of SEWAT which was used to simulate seawater intrusion in the
Gaza aquifer in vertical and horizontal sections. The results of the model predictions and
comparison were presented and discussed. A good agreement was observed between the
results of the present model in vertical cross sections and the results of SEWAT in the
horizontal cross section for the case of Gaza aquifer in terms of concentration distribution.
The construction of 2D horizontal and vertical sections can be used to obtain a 3D picture
of seawater intrusion in coastal aquifers.
5.5 Summary
In this work, a 2-D finite element model (SUFT) has been developed to study saltwater
intrusion in coastal aquifers. The developed model can handle a wide range of real SWI
problems. The developed model for simulating saltwater intrusion considers transient,
density-dependent flow and velocity-dependant dispersion. The developed finite element
model has been applied to a number of examples from the literature for validating the
model capability to simulate groundwater flow, solute transport and coupled fluid flow and
solute transport. The developed model has been validated against one of the most popular
benchmark problems in seawater intrusion in coastal aquifers, widely known as Henry’s
saltwater intrusion problem. Many researchers validated their codes against Henry’s
problem as well known cases of saltwater intrusion in order to reveal the capability of such
models to simulate saltwater intrusion in coastal aquifer. The current model has also been
applied for four different cases of Henry’s problem and gave good results for all of them
compared with the published results from other researchers. After validation of the model it
was applied to a number of real cases studies to predict seawater intrusion in coastal
aquifers in different locations of the world. A hypothetical case from the literature,
saltwater intrusion in Madras aquifer in India, saltwater intrusion in Biscayne aquifer in
Florida and saltwater intrusion in the Gaza aquifer in Palestine have been simulated by the
developed mode. The current model gave good results compared with other publications.
The results show that the current model is capable of predicting with good accuracy
saltwater intrusion in coastal aquifers.
Chapter (6) Development and application of simulation-optimization model
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Development and application of simulation-optimization model
6.1 Introduction
Management of water resources in coastal areas is very important as these areas are heavily
populated with increasing the demand for fresh water. Groundwater is considered the main
sources of water in many coastal areas. The increase of water demands increases the
withdrawal from aquifers which has resulted in lowering water tables and caused saltwater
intrusion. Saltwater intrusion is one of the main causes of groundwater quality degradation
and a major challenge in management of groundwater resources in coastal regions.
Saltwater intrusion causes an increase in chloride concentration of groundwater which
places limitations on its uses. Therefore, efficient control of seawater intrusion is very
important to protect groundwater resources from depletion.
This chapter presents development and application of a simulation-optimization (S/O)
model to control seawater intrusion in coastal aquifers using different management models.
The model is based on the integration of a genetic algorithm optimization technique and a
coupled transient density-dependent finite element model, which has been developed in this
work for simulation of seawater intrusion. The simulation model is used to compute heads
and salt concentrations in the aquifer and the optimization model is used to determine the
optimal locations, depths, abstraction/recharge rates to minimize the total cost. The
management scenarios considered include abstraction of brackish water, recharge of fresh
water and combination of abstraction and recharge. The objectives of these management
scenarios include minimizing the total costs for construction and operation, minimizing salt
concentrations in the aquifer and determining the optimal depths, locations and
abstraction/recharge rates for the wells. Also, a new methodology is presented to control
seawater intrusion in coastal aquifers. In the proposed methodology ADR (abstraction,
desalination and recharge), seawater intrusion is controlled by abstracting brackish water,
desalinating it using reverse osmosis and partially recharging it to the aquifer.
Chapter (6) Development and application of simulation-optimization model
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Development of S/O approach for the control of saltwater intrusion in coastal aquifers is a
major achievement of the present study. It marks the first time (based on the literature) that
such coupling models are used for controlling saltwater intrusion in coastal aquifers. The
developed S/O model is applied to a hypothetical case (Henry’s problem) and number of
real world case studies to control saltwater intrusion using three management models. The
efficiencies of the three management models are examined and compared. The results show
that all three models could be effective but using the ADR methodology results in the
lowest cost and salt concentration in aquifers and maximum movement of freshwater/saline
water interface towards the sea. The developed model is an effective tool to control
seawater intrusion in coastal aquifers and can be applied in areas where there is a risk of
seawater intrusion (Abd-Elhamid and Javadi, 2010).
6.2 Optimization techniques
The optimization problem can be solved through trial-and-error adjustment or through a
formal optimization technique. Numerical simulation models can be used to examine a
limited number of design options by trial-and-error. The trial-and-error method is simple
and thus widely used, testing and checking hundreds to thousands of trial solutions is
tedious and cannot guarantee that the optimal solution has been identified. In contrast, the
optimization technique can be used to search for the optimal solution in a wide search space
of design variables, and equally important, to prove whether a particular management
scenario or remedial alternative is feasible in terms of meeting the management objective
and satisfying all the constraints.
An optimization model is defined in terms of an objective function and a set of constraints.
The objective function can be formulated as the net present value of the management cost
or the total pumping rate and, if the one-time drilling and installation costs are negligible
compared to the cumulative pumping and treatment costs. The total cost includes the capital
cost of drilling and installation, operation cost associated with pumping or recharge and the
treatment costs. Groundwater management problem includes two sets of variables; decision
variables and state variables. Decision variables are the variables that used to define
different alternative decisions, such as; pumping or injection rate of a well, well location
Chapter (6) Development and application of simulation-optimization model
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and well depth. Decision variables can be managed in the calculation process to identify the
optimal management policy or strategy. State variables are the variables that describe the
flow and transport conditions of an aquifer, such as; heads and salt concentrations, which
are the dependant variables in flow and transport equations. In a coupled S/O model,
simulation components update the state variables and optimization components determine
optimal values of decision variables (Zheng and Bennett, 2002).
Coupling flow and solute transport models with optimization techniques to address
important groundwater quality management problems has started early 1980s. The
motivation for the development of S/O approach for the control of saltwater intrusion is the
enormous cost of groundwater remediation. The S/O approach has been shown to be
capable of reducing the remediation costs of contaminated land in several real-world
applications (Zheng and Bennett, 2002). The S/O approach can be used efficiently in
groundwater remediation system design and in other groundwater quality management
problems including saltwater intrusion control. The S/O approach is interesting because it
can both account for the complex behavior of a groundwater flow system and identify the
best management strategy to achieve a given set of prescribed constraints.
Optimization models are widely used in groundwater planning and management. In recent
years, a number of simulation models have been combined with optimization techniques to
address groundwater management problems. The combined simulation and optimization
model accounts for the complex behavior of the groundwater system and identifies the best
management strategy under consideration of the management objectives and constraints. In
the last two decades, genetic algorithms (GAs) have received increased attention from
academic and industrial communities for dealing with a wide range of optimization
problems. Mckinney and Lin (1994) applied a GA to groundwater resources management
and pump-and-treat system design. Ritzel et al. (1994) solved a multi-objective
groundwater contamination problem using GA. Johnson and Rogers (1995) used GA and
neural network to select the optimal well locations and pumping rates in a remediation
design problem. Zheng and Wang (2001) applied a GA to pump-and-treat system design
optimization under field conditions.
Chapter (6) Development and application of simulation-optimization model
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6.3 Genetic algorithm (GA)
GA is an optimization technique based on the process of biological evolution. GA was first
introduced by Holland (1975) and followed by Goldberg (1989), Davis (1991),
Michalewicz (1992), Mckinney and Lin (1995) and Haupt and Haupt (1998) among others.
GAs are a family of combinatorial methods that search for solutions of complex problems
using an analogy between optimization and natural selection. GA mimics biological
evolution based on the Darwinist theory of survival of the fittest, where the strongest
offspring in a generation are more likely to survive and reproduce. The GA method starts
with randomly generating an initial set of solutions, called the initial population. Each
member of this population represents a possible solution of the problem, encoded as a
chromosome. A chromosome is a string of symbols, usually a binary bit string. The
population of chromosomes evolves through a cycle that involves selection, crossover and
mutation. Each cycle is referred to as a generation. After many generations, the population
will contain chromosomes that represent near optimal solutions to the problem. GA was
applied to a wide variety of problems in engineering including groundwater management,
water resources and seawater intrusion. An overview to the application of GA in these
fields is presented in the following sections (Zheng and Bennett, 2002).
6.3.1 GA in groundwater management
GA has been applied to a number of groundwater management problems. A number of
researchers incorporated groundwater simulation models with GA to solve groundwater
management problems such as; maximizing extraction from an aquifer, minimizing the cost
of water supply, minimizing the cost of aquifer remediation and pump-and-treat. Among
the researchers who applied GA to groundwater management problems are Mckinny and
Lin (1994), Rogers and Dowla (1994), El Harrouni et al. 1996) and Aly and Peralta (1999).
Recently, GA has been used to identify cost effective solutions for pump-and-treat
groundwater remediation design. A number of researchers have applied GA to solve
groundwater pollution remediation problems (e.g. Chan Hilton and Culver 2000; Smalley et
al. 2000; Haiso and Chang 2002; Maskey et al, 2002; Gopalakrishnan et al. 2003; Espinoza
at al. 2005; Horng et al. 2005 and Chan Hilton and Culver 2005).
Chapter (6) Development and application of simulation-optimization model
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6.3.2 GA in water resources management
In the field of water resources management GA has been applied to a variety of problems
such as; calibration of rainfall-runoff models, pipe network systems and operation of
reservoirs systems. A number of researches applied GA for calibration of rainfall-runoff
models (e.g. Wang (1991); Franchini (1996) and Mulligan and Brown (1998)). A large
number of applications in pipe network optimization using GA were presented by Goldgerg
(1989), Murphy et al. (1993), Simpson et al. (1994), Halhal et al. (1997), Savic and Walters
(1997) and Farmani et al. (2005a,b). GA has also been used to optimize the operation of
reservoirs systems by Esat and Hall (1994), Oliveira and Loucks (1997) and Wardlaw and
Sharif (1999).
6.3.3 GA in seawater intrusion management
Although there have been many applications of GA in the field of water resources and
groundwater management, the applications of GA to control seawater intrusion has been
limited. For the first time in this study GA is applied to study the control of seawater
intrusion using ADR system. As the construction and operation costs of such system are
very high, thus, the optimal management has the potential to reduce costs substantially. A
number of models have been developed to analyze the control of saltwater intrusion in
coastal aquifers, based on the two different approaches; sharp interface or diffusive
interface (density-dependent). Development and application of optimization techniques to
study saltwater intrusion considering a sharp interface between freshwater and seawater
was presented by a number of researchers such as; Shamir et al. (1984) presented a multi-
objective linear programming model to relate the location of the interface toe and the
magnitude of pumping and recharge. Willis and Finney (1988) presented a planning and
management model for control of seawater intrusion in the Yun Lin regional groundwater
basin in Taiwan. The management model was formulated as a problem of optimal control
and solved using MINOS optimization model. Essaid (1990) used the finite difference
simulation model SHARP to simulate the aquifer system response within the control model.
The objective function of the model was a function of freshwater and seawater heads and
locations and magnitude of groundwater pumping or artificial recharge.
Chapter (6) Development and application of simulation-optimization model
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Finney et al. (1992) presented the development and application of a quasi three-
dimensional optimal control model for groundwater management in the Jakarta coastal
aquifer basin based on the sharp interface assumption. Hallaji and Yazicigil (1996)
presented seven groundwater management models to determine the optimal planning and
polices for a coastal aquifer in southern Turkey threatened by saltwater intrusion. They
linked steady-state and transient finite element simulation models to linear and quadratic
optimization models using response function to maximize withdrawals and minimize
pumping cost with the prevention of saltwater intrusion. Emch and Yeh (1998) developed
S/O model to manage water use within a coastal region. Two objectives are considered;
cost-effective allocation of surface water and groundwater supplies, and minimization of
saltwater intrusion. The flow model was simulated using a quasi-three-dimensional finite
difference model based on sharp interface and was linked to an optimization model,
MINOS. Tracy (1998) developed S/O model for the optimal management of Santa Barbara
city’s water resources during draught. The model linked the groundwater simulation model
MODFLOW with a linear programming model to minimize the cost of water supply subject
to water demand and hydraulic head constraints to control saltwater intrusion.
Cheng et al. (2000) presented an optimization model based on sharp interface approach to
optimize pumping in saltwater-intruded coastal aquifers. A genetic algorithm (GA) was
utilized for the search of the optimum solution. Benhachmi et al. (2001a) presented an
optimization management technique with an economic objective to maximize the benefits
from allocation of abstraction wells while minimizing the intrusion of saltwater into wells.
A sharp interface approach was incorporated into a simple genetic algorithm to search for
optimal solution. Benhachmi et al. (2001b) used a sharp-interface analytical approach to
simulate groundwater flow on a horizontal plane. The model was incorporated into a simple
genetic algorithm for optimal pumping rates and salinity control with an economic
objective. Cheng et al. (2001) used a sharp interface model to simulate saltwater intrusion
and a genetic algorithm for optimization. The objective was to maximize freshwater
extraction without causing saltwater intrusion into the wells. Moreaux and Reynaud (2001)
developed an analytical model based on sharp interface between freshwater and saltwater to
determine the optimal use of a coastal aquifer under saline intrusion. Gau and Liu (2002)
Chapter (6) Development and application of simulation-optimization model
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presented a decision model based on the loss function concept to estimate the optimum
yield from a groundwater basin and control saltwater intrusion in Yun-Lin area, Taiwan.
Zhou et al. (2003) used a quasi three-dimensional finite element model to simulate the
spatial and temporal distribution of groundwater levels in an aquifer system. They
developed a linear programming model to maximize the total groundwater pumping from
the aquifer with the control of saltwater intrusion by restricting the water levels in the
aquifer. Benhachmi et al. (2003) presented the same methodology used by Cheng et al.
(2000), with considering the economic benefit from the pumped water and minimizing
utility cost of lifting the water in the objective function. Park and Aral (2004) presented the
same methodology used by Cheng et al. (2000), but they used an iterative sub-domain
method, in which the GA searched for the optimal solution by perturbing the well locations
and pumping rates simultaneously. The numerical results obtained from the proposed
method were compared with the work of Cheng et al. (2000). The results of comparison
yielded slightly higher pumping rates than what was reported in Cheng et al. (2000).
Bhattacharjya and Datta (2005) developed S/O model for the optimal management of
saltwater intrusion in coastal aquifers. 3-D density-dependant flow and transport models
were approximated using a trained neural network to calculate salt concentration and linked
to GA-based optimization model to maximize the withdrawal of water from a coastal
aquifer, with controlling saltwater intrusion. Reichard and Johnson (2005) developed S/O
model to control saltwater intrusion in the west coast basin of coastal Los Angeles using
two management options, increased injection into barrier wells and delivery of surface
water to replace current pumpage.
On the other hand, development and application of optimization techniques in association
with saltwater intrusion considering density-dependant flow are relatively recent and very
few. Das and Datta (1999a, 1999b) presented a multi-objective management model. The
objectives were to maximize pumping from the freshwater zone and minimize pumping
from the saline zone in order to control saltwater intrusion. The nonlinear finite-difference
of the density-dependent miscible flow and salt transport model for seawater intrusion in
coastal aquifers was embedded within constraints of the management model. Das and Datta
(2000) developed a nonlinear optimization method for minimizing total sustainable yield
Chapter (6) Development and application of simulation-optimization model
172
from specific location of the aquifer while satisfying salinity constraints. Gordu et al.
(2001) developed a management model to maximize the total pumping rate from wells
considering saltwater intrusion into the aquifer. They used SUTRA code for groundwater
simulation and the general algebraic modelling system code to execute the optimization
model.
Gordon et al. (2001) developed a model for optimal management of a regional aquifer
under salinization. The source of salinization was from irrigation water percolating over
some part of the aquifer, influx of saline water from faults in the aquifer bottom, and inflow
from laterally adjacent saline water bodies. The objectives of management were to
maximize the total amount of water pumped for use and to minimize the total amount of
salt extracted with the water. The simulation model used a finite-element formulation for
the flow and a streamline upwind Petrov-Galerkin formulation for the transport. The model
computed the gradient of the state variables (heads and concentrations) with respect to the
decision variables (pumping rates at wells). The gradients were then used in a Bundle-Trust
non-smooth optimization procedure to achieve an improved solution. Rao and Rao (2004)
presented S/O approach to control saltwater intrusion through a series of extraction wells.
They used SEAWAT code to simulate saltwater intrusion and the simulated annealing
algorithm for solving the optimization problem. Two objective functions were considered:
first to maximize groundwater pumpage for beneficial purposes and second to minimize the
pumping in extraction wells which were used to extract saline water and dispose to the sea.
Park et al. (2008) investigated effects of saltwater pumping to protect an over-exploiting
freshwater pumping well. They used S/O numerical model taking into account various
parameters to identify the minimum saltwater pumping rate required to protect the over-
exploiting pumping well. Arvanitidou et al. (2010) applied a generic operational tool for the
optimal management of coastal aquifers to a real unconfined coastal aquifer in the Greek
island Kalymnos. They combined a numerical model that predicts the seawater intrusion
due to flow perturbations and the optimization method of GA. The presented management
scenarios were based on the hydrodynamic control of seawater intrusion due to the
operation of a network of productive wells and the seasonal use a of a recharge canal which
is an alternative method for increasing aquifer sustainable productivity.
Chapter (6) Development and application of simulation-optimization model
173
According to the literature, only few researchers have considered the transition zone
between the freshwater and seawater by incorporating density-dependent miscible flow and
solute transport in management of coastal aquifers. Also, the majority of the presented
management models did not consider the economic value of those systems. This work
presents the development and application of S/O model to control seawater intrusion in
coastal aquifers using three management models; abstraction of brackish water, recharge of
fresh water and combination of abstraction and recharge. The objectives of these
management models include minimizing the total costs for construction and operation,
minimizing salt concentrations in the aquifer and determining the optimal depths, locations
and abstraction/recharge rates of the wells. A new methodology is presented to control
seawater intrusion in coastal aquifers. In the proposed ADR methodology seawater
intrusion is controlled by abstracting brackish water, desalinating it using reverse osmosis
and recharging to the aquifer (Abd-Elhamid and Javadi, 2010).
6.4 Simulation-Optimization model
Simulation models have been applied for management of groundwater resources seeking
for optimal management strategy by trial-and-error. This has proved to be time consuming
and laborious. In addition, the results obtained may not be optimal. The main reason for
this is the inability of this approach to consider important physical and operational
restrictions. To accommodate these restrictions, coupling of the simulation model with a
management model is the generally adopted procedure. For groundwater systems,
combined simulation and management models may adequately predict the behavior of the
system and provide the best solution for problems such as planning of long term water
supply or preventing seawater intrusion. A relatively small number of studies have
concentrated on the control of saltwater intrusion. To the author knowledge, no attempt has
been made in the literature to develop or use S/O model to study optimal control of
seawater intrusion using a combination of abstraction and recharge wells.
Chapter (6) Development and application of simulation-optimization model
174
The proposed ADR methodology has not been used before in the control of seawater
intrusion. The only related work in the literature appears to be that of Mahesha (1996c) and
Rastogi et al. (2004) who used a simulation model to study the effect of abstraction and/or
recharge on seawater intrusion by trial-and-error. Therefore the focus of this study is on
development and application of S/O model to effectively determine the optimal solution for
control of seawater intrusion. In this work the reduction in cost is achieved by determining
the optimal depths, locations and abstraction/recharge rates while also minimizing salt
concentration in the aquifer. The main objective of this study is to determine a cost-
effective method to control the intrusion of seawater in coastal aquifers. The three main
factors contributing to the construction and operation costs are considered as the decision
variables:
1- The depth of abstraction and/or recharge wells: Reduction in depth of
abstraction/recharge wells leads to reduction in construction cost.
2- The location of well (distance from the seashore): Location of abstraction/recharge
wells can have a significant effect on reduction of the required abstraction/recharge
rates to maintain the freshwater/seawater interface in optimal position, reduction of
concentration of salt in the aquifer and reduction of operational cost.
3- Abstraction/recharge rates: Reduction of abstraction/recharge rates leads to
reduction in the operational cost.
Considering the three main factors mentioned above, the objectives of optimization are;
minimization of depths for the wells, determination of optimum locations for the wells, and
minimization of the abstraction/recharge rates subject to a number of constraints.
Appropriate weights for the objectives can be obtained by involving decision makers. In
this work minimization of the overall cost of construction and operation is considered as the
main objective. The sum of salt concentration in the region is calculated and is maintained
below a specified value or minimized. The proposed S/O model consists of two main
components. The first is a simulation model to compute hydraulic heads and salt
constructions in the aquifer. The second component is a GA-based evolutionary
optimization model that uses the results of the simulation model to compute and minimize
the objective function. The simulation model, the optimization technique and the
integration of simulation and optimization models are described below.
Chapter (6) Development and application of simulation-optimization model
175
6.4.1 Simulation model
Modelling of fluid flow and solute transport for a site involves development of a
mathematical model of the site system being studied and to use this model to predict the
value of hydraulic heads and salt concentrations at points of interest at different times.
Numerical simulation of seawater intrusion, assuming that mixing occurs at the transition
zone between seawater and freshwater, involves the solution of the partial differential
equations representing the conservation of mass for the variable-density fluid (flow
equations) and for the dissolved solute (transport equation). The governing equation and
numerical solution were presented in chapters 3 and 4. The output of the simulation model
is the pore water pressures and salt concentrations at every point within the domain. The
predicted values of pore water pressure are used to calculate pressure head and total head,
through Bernoulli’s equation.
The flow and transport equations are coupled through Darcy’s law and a constitutive
equation relating fluid density to salt concentration. This coupling between the flow and
transport equations makes the problem of seawater intrusion highly nonlinear. To simulate
the flow and transport processes, the flow and transport equations are discritized in space
and time using the finite element and finite difference techniques. The numerical solution
of coupled fluid flow and solute transport is based on solving the governing equations with
the boundary and initial conditions by the iterative solution scheme was described in
chapter 4. The flow equations are solved to compute pore water at all nodes. The pressure
head, fluid velocity and fluid density are then calculated. The calculated velocities are used
to define the dispersion coefficient for the solute transport equation. The solute transport
equation is then solved for concentrations at every node in the domain and this process is
repeated for every time step. A flow chart of the numerical solution algorithm is shown in
Figure 6.1.
Chapter (6) Development and application of simulation-optimization model
176
Figure 6.1 Flow chart of the solution algorithm
Start
Input the control parameters (i.e., domain size, dimensions)
Input boundary & initial conditions for fluid flow and solute
Input the geometric data (i.e., elements, nodes, coordinates)
Check the input data
Solve the fluid flow equation for pore water pressure
Compute heads and fluid density
Time loop
No
Store results for the next time step
Check convergence
Print the results
Last time step?
Stop No
Yes
Yes
Update model parameters for the next iteration
Compute velocity and dispersion coefficient
Solve the solute transport equation for salt concentration
Chapter (6) Development and application of simulation-optimization model
177
6.4.2 Genetic Algorithm Optimization model
Genetic algorithms (GAs) have received much attention regarding their potential as global
optimization techniques for problems with large and complex search spaces. GAs have
many advantages over the traditional optimization methods. In particular, they do not
require function derivatives and work on function evaluations alone. They have a better
possibility of locating the global optimum as they search a population of points rather than
a single point and they can allow for consideration of design spaces consisting of a mix of
continuous and discrete variables. GAs operate on a population of trial solutions that are
initially generated at random. The genetic algorithm seeks to maximize the fitness of the
population by selecting the fittest individuals, based on Darwin’s theory of survival of the
fittest, and using their genetic information in mating and mutation operations to create a
new population of solutions. Many variations of the GA have been developed, but the
algorithm implemented here can be summarized as being a simple binary encoded GA with
roulette wheel selection, uniform crossover and an elitist replacement strategy.
Saltwater intrusion control problem is formulated as an optimization problem of finding the
optimum management options in terms of depths, locations and abstraction/recharge rates
for abstraction/recharge wells to reduce the construction and operation costs and minimize
salt concentrations in the aquifer. Different sets of values for the design parameters are
generated and evolved using the principles of genetic algorithm and the behavior of each of
the generated management systems (corresponding to a set of design parameters) is
analyzed using the finite element model to evaluate its fitness in competition with other
generated systems. During successive generations the genetic algorithm operators gradually
improve the fitness of the solution and direct the search towards the optimal or near optimal
solution. GAs have some control parameters such as population size n, and probabilities for
applying genetic operators, e.g., crossover probability Pc , usually 0.7 to 1, and mutation
probability Pm , usually 0.01 to 0.1 (Qahman and Larabi 2006).
In this work, a simple genetic algorithm code written in FORTRAN is used. The selection
scheme used is tournament selection with a shuffling technique for choosing random pairs
Chapter (6) Development and application of simulation-optimization model
178
for mating. The routine includes binary coding for the individuals, jump mutation, creep
mutation, and the option for single-point or uniform crossover. The steps followed for
finding the optimal solution can be summarized as:
1- An initial population of decision variables is generated randomly in the form of
binary strings. The binary strings are then decoded to real values representing the
design parameters of the system.
2- The values of the design variables are used by the FE model to compute the
hydraulic heads and salt concentrations for different points within the domain.
3- The calculated values of heads and concentrations are used by the GA process to
evaluate the objective function (fitness) values for all members of the population.
4- The processes of selection, crossover and mutation are performed as in a simple
genetic algorithm procedure and a new population of individuals (new generation)
is created.
5- The new generation replaces the current generation and the entire process (steps 2-
4) is repeated until convergence is achieved or the maximum number of generations
is exceeded.
A flowchart for the genetic algorithm used is shown in Figure 6.2.
6.4.3 Integration of the simulation and optimization models
The simulation-optimization model developed in this work is based on the integration of a
GA with a coupled transient density-dependent FE simulation model for flow and solute
transport. In the developed simulation-optimization framework, the simulation model is
repeatedly called by the GA to calculate the response of the system to each set of design
variables generated by the GA. The simulation model is used to compute pressure heads
and salt concentrations for every node in the domain. These values for pressure heads and
salt concentrations are returned to the GA routine and used to evaluate the objective
function value and determine the fitness of the solution in competition with other generated
solutions. Figure 6.3 shows the schematic steps of the simulation-optimization model.
Chapter (6) Development and application of simulation-optimization model
179
Figure 6.2 Flowchart for the genetic algorithm
Yes
No
Evaluate new chromosomes by the FE model
Gen=Gen+1
Generate initial population and evaluate each chromosome
gen=1
Selection
Crossover
Mutation
Is Gen<max.Gen
Start
Stop
Chapter (6) Development and application of simulation-optimization model
180
Figure 6.3 Flow chart of the schematic steps of the simulation-optimization model
Calculate pressure heads and solute concentrations by
SUFT
Optimal policy
Initial parameters and data
Calculate fitness value
Update decision variable
Yes
No
Stop
Chapter (6) Development and application of simulation-optimization model
181
6.5 Formulation of the management models
An appropriate management strategy can provide an efficient and cost-effective approach
to control seawater intrusion in coastal aquifers. A GA-based optimization approach is used
in this work to incorporate a seawater intrusion simulation model within an optimization-
based management model to evolve an optimal management strategy. Three management
models are developed in this work to control seawater intrusion in coastal aquifers;
abstraction of brackish water, recharge of fresh water and combination between abstraction
and recharge. The main objective of the simulation-optimization models is to minimize the
construction and operation costs by identifying the optimal depths, locations and
abstraction/recharge rates for abstraction and/or recharge wells to control the intrusion of
seawater. Figure 6.4 shows the design variables considered in the simulation-optimization
model.
Figure 6.4 Schematic sketch for potential locations and depths for the abstraction and
recharge wells
C=0.5
L
D LA
LR
QR
DR
QA
DA
Chapter (6) Development and application of simulation-optimization model
182
6.5.1 Management model 1: (Abstraction wells)
Management of coastal aquifers often requires consideration of multiple objectives. One
management option could be to abstract brackish water from the saline zone to decrease the
volume of saline water and retard the position of the freshwater-seawater interface.
However, abstraction of brackish water from a coastal aquifer, in some cases, may increase
the intrusion of seawater. Therefore, the management objective should also minimize the
total amount of salt in the aquifer. The objective of model 1 is to minimize the total
construction and operation costs of abstraction wells and to minimize the total amount of
salt in the aquifer. This can be achieved by determining the optimum depth, location and
the abstraction rate of abstraction well to reduce the total cost. The decision variables are
therefore location, depth and abstraction rate and the state variables are; pressure heads and
salt concentrations at every node in the domain that are computed by the simulation model.
The objective functions of model 1 can be represented mathematically as follows:
Min N
iicf
11 (6.1)
Min N
iTAA CCQf
i12 )( (6.2)
Min )(1
3 DW
N
iA CDf
i (6.3)
where
f : is the objective function in terms of the total cost
N : is the total number of nodes in the domain
DA : is depth of abstraction well (m)
QA : is the abstraction rate (m3/sec)
c : is total amount of solute mass in the aquifer (mg/l)
CA : is the cost of abstraction ($/m3)
CT : is the cost of treatment ($/m3)
CDW : is the cost of installation/drilling of well ($/m)
Chapter (6) Development and application of simulation-optimization model
183
To deal with multiple objective functions the weighted sum of each objective function can
be used to form a single scalar objective function (Park and Aral, 2004). This brings the
difficulty of adjusting the weight of each objective, which can be considered to be a
management decision. However, it is recognized that a single scalar objective function
generated is not capable of representing the vector tendency of each objective. Nonetheless,
in this work the single scalar objective function approach is used for simplicity. This
approach was used by many researchers such as Gordon (2001) Park and Aral (2004) and
Qahman and Larabi (2006). The final objective functions of model 1 can be represented as:
Min )()(1
31
21
1 DW
N
iA
N
iTAA
N
ii CDPCCQPcPf
ii (6.4)
where P1, P2 and P3 are the weighting parameters, which are chosen based on the literature.
The management objective must be achieved within a set of constraints, which can be
derived from technical, economic, legal or political considerations. The constraints in this
model include; well depth, well location and abstraction rate and can be expressed as:
maxmin AAA QQQi
(6.5)
maxmin AAA LLLi
(6.6)
maxmin AAA DDDi
(6.7)
Equations 6.5-6.7 represent side constraints, where minAQ ,
maxAQ are the minimum and
maximum values of abstraction rate, minAL ,
maxAL are the minimum and maximum distances
from the abstraction well to the sea shore and minAD and
maxAD are the minimum and
maximum depths of the abstraction well. In this management model the costs of
installation/drilling of well, abstraction and treatment (i.e. desalination) are introduced
based on the literature. According to the literature these costs are considered as; cost of
Chapter (6) Development and application of simulation-optimization model
184
installation/drilling of well per unit depth: $1000, cost of abstraction per cubic meter: $0.42
and cost of treatment (i.e. desalination) per cubic meter: $0.6 (Qahman and Larabi, 2006).
6.5.2 Management model 2: (Recharge wells)
The second management model is designed to search for the optimal arrangement for
recharge of fresh water into the aquifer to control seawater intrusion. The management
model involves recharge of freshwater to increase the volume of freshwater in the aquifer to
retard the position of the freshwater/seawater interface. However, recharge of fresh water
may increase the total cost required to prevent seawater intrusion due to the large amount of
water required for recharge and the availability and cost of fresh water in coastal areas.
Therefore, this management model is used to minimize the cost of water required to be
recharged. The objective of model 2 is to minimize the total construction and operation
costs of recharge wells and to minimize the total amount of salt in the aquifer. Location,
depth and recharge rate are considered as the decision variables to be optimized to reduce
the total cost. A single scalar objective function is used by combining of a number of
objectives, following a procedure similar to the one adopted in the management model 1.
The objective function of model 2 can be represented as:
Min )()(1
51
41
1 DW
N
iR
N
iRPWR
N
ii CDPCCQPcPf
ii (6.8)
Where
f : is the management objective function in terms of the total cost
P4 and P5: are the weighting parameters
QR : is the recharge rate (m3/sec)
DR : is the depth of recharge well (m)
CR : is the cost of recharge ($/m3)
CPW : is the price of water ($/m3)
The constraints of model 2 can be expressed as:
Chapter (6) Development and application of simulation-optimization model
185
maxmin RRR QQQi
(6.9)
maxmin RRR LLLi
(6.10)
maxmin RRiR DDD (6.11)
Equation (6.9) constrains the recharge rate of an recharge well between specified minimum
and maximum values minRQ ,
maxRQ . Equation (6.10) is a constraint on the distance of
recharge well from the seashore between specified minimum and maximum values minRL ,
maxRL . Equation (6.11) is a constraint on the depth of recharge well between specified
minimum and maximum values minRD and
maxRD . In management model 2 the costs of
installation/drilling of well, recharge and price of water including transportation is
introduced based on the available data from the literature. These costs are assumed to be as
follows; cost of installation/drilling of well per unit depth: $1000, cost of recharge per
cubic meter: $0.48 and price of water per cubic meter: $1.5 (Qahman and Larabi, 2006).
6.5.3 Management model 3: (Abstraction and Recharge wells)
Management model 3 is developed in order to combine management models 1 and 2 to
prevent/control seawater intrusion. The objective of management model 3 is to minimize
the total construction and operation costs of abstraction and recharge wells and to minimize
the total amount of salt in the aquifer. Locations, depths and abstraction/recharge rates of
the abstraction and recharge wells are considered as decision variables.
The objective function of model 3 can be represented as:
Min )()(1
31
21
1 DW
N
iA
N
iTAA
N
ii CDPCCQPcPf
ii
)()(1
51
4 DW
N
iR
N
iRR CDPCQP
ii (6.12)
Chapter (6) Development and application of simulation-optimization model
186
The same constraints as in equations (6.5, 6.6, 6.7, 6.9, 6.10, and 6.11) are used in
management model 3.
The design of optimal ADR system aims to determine the optimal locations, depths and
abstraction/recharge rates of the abstraction and/or recharge wells, to minimize the total
cost of construction and operation of abstraction and recharge wells and treatment of
abstracted water, and to minimize the total amount of salt in the aquifer. Management
model 3 is used to apply the new methodology ADR to control seawater intrusion in order
to get the optimal design parameters. The results of applying the three management models
are presented and discussed in the following section. Also, a comparison is presented
between the three models in terms of total cost, total amount of water abstracted or
recharged and total salt concentration in the aquifer.
6.6 Application of the simulation-optimization model
This section presents the application of the developed simulation-optimization model to
control seawater intrusion in coastal aquifers. The objective of the management model is to
minimize the total cost of construction and operation to control saltwater intrusion in
coastal aquifers. The three management models are applied to a hypothetical coastal aquifer
(Henry’s problem) and a real world case study, Biscayne aquifer, Florida, USA.
6.6.1 A hypothetical case study (Henry’s problem)
This example presents the application of the developed simulation-optimization model to
control seawater intrusion in a hypothetical confined coastal aquifer. This problem is
known as Henry’s problem (Henry, 1964). The geometry and initial and boundary
conditions are shown schematically in Figure 5.7 (chapter 5). The dimensions of the aquifer
are considered to be 200 m (length) and 100 m (depth). The finite element mesh consists of
661 nodes and 200 elements, each of size 10.0 m by 10.0 m. The simulation parameters for
the problem are given in Table 5.1 (chapter 5). The results of the finite element simulation
(without optimization) are shown in Figure 6.5 which shows 0.5 isochlor in the aquifer.
Chapter (6) Development and application of simulation-optimization model
187
Figure 6.5 0.5 isochlor for the hypothetical case study
Potential locations and depths for the abstraction and recharge wells are shown in Figure
6.6.
Figure 6.6 Schematic sketch for potential locations and depths for the abstraction and
recharge wells of the hypothetical case study
200.0
100.0
QA QR
LR
LA
DA DR
Chapter (6) Development and application of simulation-optimization model
188
In the developed S/O framework, initially a population of sets of design variables
(locations, depths and abstraction and/or recharge rates for the wells) is generated randomly
by the GA. The FE model is then called to evaluate the response of the system to each set
of design variables in the population in the form of heads and salt concentrations
throughout the domain. The computed values for pressure heads and salt concentrations are
returned to the GA routine and used to evaluate the objective function value and determine
the fitness of each solution in competition with other generated solutions. The population
then undergoes the genetic operations of selection, cross over and mutation to create a new
population. The parameters used by GA are: population size = 10, maximum generations
=100, the values of crossover and mutation probabilities are 0.7 and 0.03 respectively. The
upper and lower limits for the side constraints are listed in Table 6.1.
Table 6.1 Summary of the upper and lower constraints for the hypothetical case study
Parameter value
Qmin (m3/s) 0.0
Qmax (m3/s) 0.1
Lmin (m) 0.0
Lmax (m) 200.0
Dmin (m) 0.0
Dmax (m) 100.0
The application of the proposed simulation-optimization model to the current case has been
done in two steps. In the first step, the simulation model is applied with the initial and
boundary conditions to compute heads and salt concentrations at every node in the domain.
In the second step, the simulation-optimization model is applied to determine the decision
variables; well depths, locations and abstraction/recharge rates subject to constraints which
have been defined for the three management models presented above. The objective
function for the three models is to minimize the total cost associated with well locations,
depths and abstraction/recharge rates. The results obtained from the simulation-
optimization model for the three management models are presented in Table 6.3 and
discussed in the following section.
Chapter (6) Development and application of simulation-optimization model
189
6.6.1.1 Application of management model 1
In management model 1 the objective is to minimize the total construction and operation
costs of using abstraction wells to control seawater intrusion. The construction costs
include installation and drilling costs, while operation costs include abstraction and
treatment (i.e. desalination) costs. The values of the costs used in this case are listed in
Table 6.2 (Qahman and Larabi, 2006).
Table 6.2 Summary of the values of costs
Parameter Value
CA ($/m3) 0.42
CT ($/m3) 0.6
CDW ($/m) 1000.0
CR ($/m3) 0.48
CPW ($/m3) 1.5
The optimization problem for management model 1, as defined in equation (6.4) and
subject to the constraints defined by equations (6.5), (6.6) and (6.7), is solved using the
genetic algorithm to obtain the optimum depth, location, abstraction rate to minimize the
cost of construction and operation of the system and the salt concentrations in the aquifer.
The results in terms of the optimal depth, location and abstraction rate for the well, together
with the corresponding total cost are summarized in Table 6.3. Using management model 1
the value of the total cost is determined as $2.62 million per year. The optimal depth is 90
m, the optimal location is 50 m from the seashore and the optimal abstraction rate is 0.083
(m3/sec) while the total concentration in the aquifer is reduced from 167 to 149 (mg/l). The
final optimized concentration distribution for this model is shown in Figure 6.7 in the form
of 0.5 isochlor. Figure 6.7 shows how the distribution of salt and the interface between
saline water and fresh water are affected by the current scenario of control.
Chapter (6) Development and application of simulation-optimization model
190
6.6.1.2 Application of management model 2
In the second management model the objective is to determine the optimal well location,
depth and recharge rate to minimize the total construction and operation costs of using
recharge wells to control seawater intrusion. The optimization problem for management
model 2, as defined in equation (6.8) subject to the constraints defined in equations (6.9),
(6.10) and (6.11) is solved using the GA to obtain the optimum depth, location, recharge
rate and minimum cost of constructing and operating the system. The optimized well depth,
location and recharge rate in addition to the optimal total cost are listed in Table 6.3. Using
management model 2 the total cost is $5.72 million per year. The optimal depth is 60 m, the
optimal location is 90 m from the seashore and the optimal recharge rate is 0.095 (m3/sec)
while the total concentration has reduced from 167 to 151 (mg/l). The final optimized
concentration distribution for this model is shown in Figure 6.7.
6.6.1.3 Application of management model 3
The third management model is based on combination of management models 1 and 2. This
model seeks to minimize the total construction and operation costs of using both abstraction
and recharge wells to control seawater intrusion. The optimization problem for
management model 3, as defined in equation (6.12) and the constraints used in management
models 1 and 2 is solved to obtain the optimum depths, locations, abstraction/recharge rates
for both abstraction and recharge wells to minimize the cost of construction and operation
of the system. The results in terms of the optimal depths, locations and abstraction and
recharge rates in addition to the corresponding total cost are listed in Table 6.3. Using
management model 3 the value of the minimized objective function is $1.32 million per
year. The optimal depths for abstraction and recharge wells are 90 and 80 m respectively;
the optimal locations for abstraction and recharge wells are 50 and 110 m from the seashore
and the optimal rates for abstraction and recharge wells are 0.048 and 0.018 (m3/sec)
respectively. The total concentration in the aquifer has reduced from 167 to 142 (mg/l). The
final optimized concentration distribution for this model is shown in Figure 6.7.
Chapter (6) Development and application of simulation-optimization model
191
Table 6.3 Summary of the results obtained from the simulation-optimization models for the
hypothetical case study
Model Norm.
L
Norm.
D
Q
(m3/ s)
Norm.
C
Q
(Mm3/ year)
F
(cost $/year)
No management
model
No abstraction or recharge
wells have been used
167
Abstraction well
A
50
90
0.083
149
2.6
2.62E+6
Recharge well
R
90
60
0.095
151
3.0
5.72E+6
Abstraction and
Recharge well
A
R
50
110
90
80
0.048
0.018
142
142
1.5
0.5
1.32E+6
Figure 6.7 0.5 isochlor distribution from simulation-optimization models for the
hypothetical case
Chapter (6) Development and application of simulation-optimization model
192
6.6.1.4 Discussion
This case presented the application of simulation-optimization model to control seawater
intrusion in a hypothetical costal aquifer. Three management models with different
objective functions were considered. The results of the simulation-optimization model for
the three management scenarios were presented and compared. The results show that the
developed model is capable of determining the optimal depths, locations and abstraction or
recharge rates for the wells and minimizing the total cost of construction and operation.
Comparison between total costs, salt concentrations in the aquifer and abstraction/recharge
rates for the three management models are presented in Figures 6.8, 6.9 and 6.10. From
these figures it can be concluded that:
1- Using recharge wells alone, requires a higher cost of $5.72 million per year mainly
due to the price of water and its transportation. Using abstraction wells alone requires
a lower cost of $2.62 million per year. However, using a combination of abstraction,
treatment (i.e. desalination) and recharge results in the lowest cost of $1.32 million
per year which represents 50% of the abstraction costs and 25% of the recharge costs.
2- Using the management model 2 involving recharge wells only, requires recharge of 3
(Mm3/year) of fresh water. Using abstraction wells (management model 1) requires
abstraction of 2.6 (Mm3/year) of saline water. However, using model 3 requires
abstraction of 1.5 (Mm3/year) of saline water and recharge of 0.5 (Mm3/year) of fresh
water which is less than the total amount of water abstracted or recharged in
management models 1 or 2.
3- Using the management model 2 (recharge wells only), reduced the total salt
concentration in the aquifer from 167 to 151(mg/l) and using abstraction wells
(management model 1) reduced the salt concentration to 149 (mg/l). However, using
model 3 reduced salt concentration to 142 (mg/l) which is less than the amount of salt
concentration achieved using management models 1 and 2.
4- The third model gave the lowest cost partly because the cost associated with the
supply of water used for recharge does not apply in this case as the required water is
provided primarily from the desalination plant. In addition the excess treated water
can be directly used for other purposes. Management model 3 appears to be very
Chapter (6) Development and application of simulation-optimization model
193
efficient and more practical, as it requires substantially lower cost and leads to further
advancement of the freshwater-seawater interface towards the sea and lower
concentration of salt in the aquifer.
Figure 6.8 Comparison between total costs of model 1, 2 and 3
for the hypothetical case study
Figure 6.9 Comparison between total salt concentrations remaining in the aquifer
of model 1, 2 and 3 for the hypothetical case study
Chapter (6) Development and application of simulation-optimization model
194
Figure 6.10 Comparison between total abstraction/recharge rates
of model 1, 2 and 3 for the hypothetical case study
6.6.2 Real world case study (Biscayne aquifer, Florida, USA)
In this case the developed simulation-optimization model is applied to a real case study to
control saltwater intrusion in a coastal aquifer. The real world problem is selected from the
coastal area of Biscayne aquifer at Coulter area, Florida, USA. The aquifer average depth is
30.0 m below mean sea level and the length is 300.0 m. The domain is discretized using
two dimensional grid ( X= Y=6 m). The mesh consists of 250 elements and 861 nodes.
Coefficient of permeability of 3x10-10 m/sec and porosity of 0.39 were considered. The
longitudinal and transverse dispersivities L and T were taken as 5.0 and 0.5 m,
respectively. The density of fresh water f and seawater s are 1.0 and 1.025 t/m3
respectively. The schematic sketch of the Biscayne aquifer and the corresponding boundary
conditions applied in this case are shown in Figure 5.20 (chapter 5). The Biscayne aquifer
is simulated using the developed finite element code and the results of simulation, without
optimization are shown in Figure 6.11 in the form of 0.25, 0.5, 0.75 and 1.0 isochlor
contours.
Chapter (6) Development and application of simulation-optimization model
195
Figure 6.11 Isochlor counters in Biscayne aquifer, Florida
Potential locations and depths for the abstraction and recharge wells are shown in Figure
6.12.
Figure 6.12 Schematic sketch for potential locations and depths for the abstraction and
recharge wells of Biscayne aquifer, Florida
300.0
30.0
QA QR
LR
LA
DA DR
Chapter (6) Development and application of simulation-optimization model
196
The GA is applied using the following parameters: population size = 10, maximum
generations =100, the values of crossover and mutation probabilities are 0.7 and 0.03
respectively. The values of the costs used in this example according to the literature are
presented in Table 6.2. The upper and lower limits for depths, locations and
abstraction/recharge rates are listed in Table 6.4.
Table 6.4 Summary of the upper and lower of constraints for Biscayne aquifer, Florida
Parameter Value
Qmin (m3/s) 0.0
Qmax (m3/s) 0.01
Lmin (m) 0.0
Lmax (m) 300.0
Dmin (m) 0.0
Dmax (m) 30.0
The simulation model is applied with the initial and boundary conditions to compute
pressure head and salt concentration at every node in the interred domain. The results of the
finite element simulation (with no optimization) are shown Figure 6.11 in the form of
isochlor contours in the aquifer. The simulation-optimization model is then applied to
minimize the costs and obtain the optimal depths, locations and abstraction/recharge rates
for the wells. The final optimized concentrations distribution for this example is
summarized in Figure 6.13. The results obtained from the simulation-optimization model
for the three management models are presented in Table 6.5 and discussed in the following
sections.
Chapter (6) Development and application of simulation-optimization model
197
6.6.2.1 Application of management model 1 Management model 1 is applied to minimize the total construction and operation costs of
using abstraction wells. The optimization problem for management model 1 aims to
determine the optimum depth, location, abstraction rate to minimize the cost of constructing
and operating the system. The optimized well depth, location and abstraction rate in
addition to the optimal cost are listed in Table 6.5. Using management model 1, the value
of total cost is determined as $0.25 million per year. The optimal depth is 24 m, the optimal
location is 66 m from the sea shore and the optimal abstraction rate is 0.008 (m3/sec) while
the total concentration in the aquifer has reduced from 305 to 248 (mg/l).
6.6.2.2 Application of management model 2
The objective of the second management model is to determine the optimal well location,
depth and recharge rate to minimize the total construction and operation costs of using
recharge wells to control saltwater intrusion. The optimized well depth, location and
recharge rate and the optimal total cost are presented in Table 6.5. Using management
model 2 the total cost is $0.5 million per year. The optimal depth is 24 m, the optimal
location is 156 m from the sea shore and the optimal abstraction rate is 0.009 (m3/sec)
while the total concentration in the aquifer has reduced from 305 to 255 (mg/l).
6.6.2.3 Application of management model 3
The management model 3 which combines models 1 and 2 aims to minimize the total
construction and operation costs of using both abstraction and recharge wells to control
saltwater intrusion. The optimized well depths, locations and abstraction and recharge rates
and the optimal total cost are listed in Table 6.5. Using management model 3 the total cost
is $0.13 million per year. The optimal depths for abstraction and recharge wells are 24 and
18 m, respectively. The optimal locations for abstraction and recharge wells are 72 and 186
m respectively from the sea shore and the optimal rates for abstraction and recharge wells
are 0.005 and 0.002 (m3/sec), respectively while the total concentration in the aquifer has
reduced from 305 to 227 (mg/l).
Chapter (6) Development and application of simulation-optimization model
198
Table 6.5 Summary of the results obtained from the simulation-optimization models for
Biscayne aquifer, Florida
Model Norm.
L
Norm.
D
Q
(m3/ s)
Norm.
C
Q
(Mm3/ year)
F
(cost $/year)
No management
model
No abstraction or recharge
wells have been used
305
Abstraction well
A
66
24
0.008
248
0.252
0.25 E+6
Recharge well
R
156
24
0.009
255
0.283
0.5 E+6
Abstraction and
Recharge well
A
R
72
186
24
18
0.005
0.002
227
227
0.158
0.063
0.13 E+6
Figure 6.13 0.5 isochlor distribution from simulation-optimization models for Biscayne
aquifer, Florida
Chapter (6) Development and application of simulation-optimization model
199
6.6.2.4 Discussion
In this case the application of the simulation-optimization model to control saltwater
intrusion to a real world case study using the three management models is presented. The
results are presented in Table 6.5. Comparisons between total costs, total salt
concentrations in the aquifer and total abstraction/recharge rates are presented in Figures
6.14, 6.15 and 6.16.
Figure 6.14 Comparison between total costs of model 1, 2 and 3
of Biscayne aquifer, Florida
Figure 6.15 Comparison between total salt concentrations remaining in the aquifer
of model 1, 2 and 3 for Biscayne aquifer, Florida
Chapter (6) Development and application of simulation-optimization model
200
Figure 6.16 Comparison between total abstraction/recharge rates
of model 1, 2 and 3 for Biscayne aquifer, Florida
From the above comparison charts it can be noticed that using the proposed system ADR in
model 3 reduced salt concentration from 305 to 227 (mg/l) which is less than the amount of
salt concentration reduced by model 1 (248 mg/l) or 2 (255mg/l) furthermore, using ADR
requires abstraction of 0.158 (Mm3/year) of saline water and recharge of 0.063 (Mm3/year)
of fresh water which is less than the amount of water abstracted using model 1 (0.252
Mm3/year) or recharged using model (0.283 Mm3/year). It is also gives the lowest cost
($0.13 million per year) compared with the cost for recharge ($0.5 million per year) and
abstraction ($0.25 million per year) which represents 50% of the abstraction costs and 25%
of the recharge costs.
6.7 Summary
A simulation-optimization model was developed to study the control of seawater intrusion
in coastal aquifers using three management models; abstraction of brackish water, recharge
of fresh water and combination of abstraction and recharge. The developed model was
applied to evaluate the three management scenarios to control seawater intrusion in coastal
aquifers and to determine the optimal locations, depths and rates of abstraction and/or
recharge wells in addition to minimizing the construction and operation costs. The
Chapter (6) Development and application of simulation-optimization model
201
efficiencies of the proposed management scenarios have been examined and compared. The
results show that all three scenarios could be effective in controlling seawater intrusion but
using model 3; i.e., ADR methodology, resulted in the lowest cost and salt concentration in
aquifers and maximum movement of freshwater/saline water interface towards the sea. The
results also show that for the case studies considered in this work, the amount of abstracted
and treated water is three times the amount required for recharge; therefore the remaining
treated water can be used directly for different proposes. The developed model is an
effective tool to control seawater intrusion in coastal aquifers and can be applied in areas
where there is a risk of seawater intrusion. The application of ADR methodology through
the third model appears to be more efficient and more practical, since it is a cost-effective
method to control seawater intrusion. Therefore, it can be used for sustainable development
of water resources in coastal areas where it provides a new source of water that comes from
treated water.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
202
Effect of climate change and sea level rise on saltwater intrusion
7.1 Introduction
Climate change is increasing at an alarming rate due to human activities and natural
processes. Scientists worry that human societies and natural ecosystems might not adapt to
rapid climate change. Majority of climatologists believe that human activities are
responsible for most of the climate change by increasing the emission of greenhouse gases
that increase the temperature of the earth. A small number of scientists state that natural
processes could have caused global warming. Those processes include increase in the
energy emitted by the sun. But the vast majority of climatologists believe that increases in
the sun's energy have contributed only slightly to recent warming. Continued climate
change might, over centuries, melt large amounts of ice from glaciers, caps and sheet. As a
result, the sea levels would rise throughout the world. Many coastal areas would experience
flooding, erosion, loss of wetlands, and entry of seawater into fresh groundwater aquifers.
High sea levels would submerge some coastal cities, small island nations, and other
inhabited regions.
Climate change, sea level rise and saltwater intrusion present the future challenge of water
resources management in coastal areas. Scientists expect that the climate change will
continue over the current century in spite of the international efforts to reduce greenhouse
gases emission. This change is expected to exacerbate already existing environmental
problems in many countries. In particular, coastal areas all over the world are expected to
suffer from impacts of sea level rise (SLR), as well as other impacts, in addition to already
existing problems of coastal erosion, subsidence, pollution, land use pressure and
deterioration of ecosystems. Global mean sea-level rise has been estimated between 1.0-2.0
mm/yr during the last century (IPCC, 1996). Future sea-level rise due to atmospheric
climate change is expected to occur at a rate greatly exceeding that of the recent past. By
2100 it is expected that the rise in sea levels would be between 20 cm to 88 cm (IPCC,
2001).
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
203
The most important direct effects of sea level rise on coastal regions are; coastal erosion,
shoreline inundation and saltwater intrusion into coastal aquifers. The problem is quite
serious and presents the present and future challenges for all interested in this subject
including; hydrologists, environmental engineers and civil/geotechnical engineers. The
solution of this problem is not easy and requires many organizations all over the world to
work together to face the incoming disaster. This chapter reviews the causes of climate
change and their effects on coastal areas. Also, the effects of sea level rise on saltwater
intrusion and methods of controlling erosion, inundation and saltwater intrusion are
discussed. The final section of the chapter presents the application of the developed model
(SUFT) to study the effects of likely climate change and sea level rise on saltwater
intrusion in coastal aquifers.
7.2 Climate change
Climate change is a result of natural and/or man-made activities. The average temperature
of the globe has changed over the past century due to the increase in concentrations of
greenhouse gases mainly carbon dioxide (CO2), nitrous oxide (N2O) and methane (CH4).
The main human activities that contribute to global warming are the burning of fossil fuels
such as coal, oil, and natural gas. Most of the burning occurs in automobiles, in factories,
and electric power plants that provide energy for houses and office buildings. The burning
of fossil fuels creates greenhouse gases that slow the escape of heat to space. Greenhouse
gases allow solar radiation to pass through the atmosphere to the earth surface, but they
intercept and store the infrared radiation emitted from the earth surface. This leads to
warming of the atmosphere. This phenomenon is known as global warming which is
considered the main cause of climate change. The increase in concentration of greenhouse
gases is associated with a corresponding increase in the average global temperature which
is expected to be higher by the end of the current century. Measurements showed that the
average temperature of the earth has risen by 0.5-0.7o C, during the last century. A
significant increase in the concentration of greenhouse gases is expected in the current
century. Studies indicated an increase of average global temperature of the earth by 1.4-5.8o
C by the end of this century.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
204
Figure 7.1 shows different scenarios for estimating the future change in global temperature
of earth from 1900 to 2100 (IPCC, 2001).
Figure 7.1 Scenarios for estimating average global temperature of earth (IPCC, 2001)
Continued global warming could have many damaging effects. It might harm plants and
animals that live in the sea. It could also force animals and plants on land to move to new
habitats. Weather patterns could change, causing flooding, drought, and an increase in
damaging storms. Global warming could melt enough polar ice to raise the sea level. In
certain parts of the world, human disease could spread and crop yields could decline.
Global warming and climate change have a serious consequence on hydrologic cycle
components as it changes the quantity and quality of the components in space, time and
frequently domain (Sherif et al, 1999). Therefore, the increase of greenhouse gases
emission should be controlled to protect the earth and its inhabitants from these serious
changes.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
205
7.3 Sea level rise
Residents of coastal areas are becoming increasingly concerned about the effects of climate
change as well as sea level rise because of possible higher rates of erosion, inundation and
saltwater intrusion. Climate change also has a great effect on coastal aquifers which are
subject to saltwater intrusion because it leads to the increase of mean sea level which has a
direct effect on saltwater intrusion. The increase in global temperature will warm the land
surface, oceans and seas. This warming will decrease the atmospheric pressure which will
in turn lead to the increase of water level in the oceans and seas. The rise in seas and oceans
water levels is for a number of reasons including:
1- Thermal expansion of oceans and seas,
2- Melting of glaciers and ice caps and
3- Greenland and Antarctic ice sheets melting.
Climate change has already caused changes in the sea levels during the last decades.
Computer models help climatologists to analyze past climate changes and predict future
changes. Scientists use climate models to describe how various factors affect the
temperature of Earth's surface. Such models help climatologists to estimate the future
changes in temperature and sea levels. The global mean sea-level has risen by 10-20 cm
over the past century as estimated by Intergovernmental Panel of Climate Change (IPCC
1996). Future sea-level rise is expected to occur at a rate greatly exceeding that of the
recent past; the predicted increase in the sea level rise is in the range of 20-88 cm within the
next 100 years (IPCC 2001). A number of studies have been carried out to estimate the
future sea level changes based on past sea level changes due to thermal expansion, glaciers
and ice caps and Greenland and Antarctic ice sheets melting. Figure 7.2 shows the
estimated global average sea level rise between 1990 and 2100 (IPCC, 2001).
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
206
Figure 7.2 Scenarios for estimating global average sea level rise (IPCC, 2001).
7.3.1 Effects of sea level rise on coastal areas
Continued climate change and sea level rise could have many effects on the coastal areas
on the long term including (Canning, 2001):
1- increase of coastal erosion,
2- coastal inundation of unprotected low lying areas,
3- subsidence of shore cities which lie in low land,
4- change in the tidal prism,
5- seawater intrusion,
6- coastal water table rise which has led to;
- agricultural soil saturation,
- long duration of flooding,
- impendence of on-site sewage disposal and
- corrosion of underground pipes and tanks.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
207
The coastal erosion may lead to partial or full submergence of some coastal cities which lie
in low land. Sea level rise also threatens the groundwater storage in coastal aquifers
because the sea level rise imposes an additional pressure head at the seaside which helps to
move saltwater into the aquifer. The rise in sea levels will shift the mixing zone further
inland. The extraction wells that are located in fresh groundwater may then be located in
saline water as shown in Figure 7.3 and consequently, the abstraction rates of these wells
may be reduced or the wells may be abandoned. This is considered one of the most serious
impacts of sea level rise on coastal regions. In addition, crops will suffer from salt damage
due to the increased salinity of the soil, which could have serious economic effects.
Figure 7.3 Effect of sea level rise on abstraction wells
The effects of climate change and sea level rise on saltwater intrusion in the long term
should be considered and controlled because few centimeters rise in sea levels could have a
great effect on saltwater intrusion and help to shift the transition zone more inland. Some
investigations have been done to study the likely effect of climate change and sea level rise
on saltwater intrusion in different locations of the world. For example, in the Nile Delta
aquifer in Egypt investigations have shown that saltwater intrusion is very vulnerable to
climate change and sea level rise. It is expected that some shore cities will be submerged
and the saltwater will intrude kilometers inside the Delta by the end of the current century
due to the increase in the Mediterranean Sea level. Also saltwater intrusion has serious
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
208
effects on coastal cities as it leads to the contamination and depletion of groundwater. For
example, in Gaza strip the main source of water is groundwater and the population growth
rate is excessively high. The shortage of water has resulted in over-pumping from the
aquifer which has lead to dramatic increase in the intrusion of saltwater. With the
continuous population growth and over-pumping it is expected that during the next decade
the aquifer will be completely polluted by saline water and inhabitants should find a new
source of water. Therefore, the control of saltwater intrusion has become essential in these
areas in order to protect the groundwater resources. The next section discusses the effects
of sea level rise on erosion, inundation and saltwater intrusion.
7.3.1.1 Erosion and inundation
Climate change and sea level rise may result in extensive shore erosion and shoreline
inundation. Sand transport on beaches is controlled by wave action which can move sand in
both onshore and offshore directions. Waves break the beach profile and cut the beach face
back, depositing the sand offshore. Thus, the erosion of beaches during an essentially static
sea level is caused by waves carrying sand offshore. With a significant rise in sea levels
there will be an acceleration of beach erosion because:
1- The higher water level allows waves to act further up on the beach which results in
more erosion;
2- At higher sea levels, waves can get closer to the shore before breaking and cause
increased erosion;
3- Deeper water also decreases wave refraction and thus increases the capacity of
alongshore transport;
4- Higher sea levels could change the source or sediments by decreasing river transport
to the sea as the river mouth is flooded.
A rise of sea levels will cause coastal inundation, an effect that is difficult to separate from
the effect of shore erosion where erosion is occurring. At the water line, typical beach
profile have a slope that can vary from 1:5 to 1:100, so a 1 cm sea level rise can move the
shoreline from 5 to 100 cm inland in addition to landward movement of the profile owing
to erosion. Deeper coastal water caused by sea level rise will increase the tidal range along
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
209
the coast, which will increase the effects on inundation. Also, submergence of low level
cities will increase with the increase in sea levels in different locations of the world.
7.3.1.2 Saltwater intrusion
Saltwater intrusion is a major problem in coastal regions as it increases the salinity of
groundwater and water may become unsuitable for human use. Salinization of groundwater
is considered a special category of pollution that threatens groundwater resources, because
mixing a small quantity of saltwater in the groundwater makes freshwater unsuitable and
can result in abandonment of freshwater supply. High population growth rate in the coastal
regions where about two third of the world population live has increased water demand;
this has lead to excessive abstraction from the aquifers which has resulted in the migration
of saltwater towards the aquifers. Over-pumping results in decrease of water table or
piezometric head and hence more intrusion will occur. Furthermore, the rise in sea levels
accelerates the saltwater intrusion into the aquifers because it imposes an additional
pressure head on the sea side of the aquifer.
The relationship between sea level rise and saltwater intrusion can be approximately
estimated according to the Ghyben-Herzberg relationship, which states that the depth of
interface below mean sea level equals to 40 times the height of the potentiometric surface
above the mean sea level. Therefore, a one meter increase in sea level rise could cause a 40
m reduction of freshwater thickness as shown in Figure 7.4. With the impact of sea level
rise combined with over-pumping the problem becomes very serious and requires practical
measures to protect the available water resources from pollution.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
210
Figure 7.4 The relationship between sea level rise and saltwater intrusion
7.3.2 Methods of controlling the effects of sea level rise on coastal areas
This section presents specific methods of controlling coastal erosion, shoreline inundation
and saltwater intrusion, which is caused by sea level rise.
7.3.2.1 Control of erosion and inundation
To prevent shoreline erosion and inundation one must be keep waves from attacking the
shore by intersecting them seaward of the surf zone or by armoring that portion of the shore
profile where erosion occurs. Erosion can be prevented by reducing the ability of waves to
transport sand or by increasing the supply of sand available for transport by waves so that
the sand is not removed from the beach to satisfy the sand transport capacity. In coastal
regions, inundation caused by the rise in sea level and the increase in tide range can be
prevented by constructing a water-tight continuous structure such as dike or retaining
structure. Increased wave attack owing to higher water levels or water attack at higher
elevations because of the raised sea levels will require many coastal structures to be
stabilized to suite the new levels of water. Higher sea levels will diminish the functionality
Water table
Saline groundwater
Fresh groundwater Sea
Sea level
Ground surface
40.0 m
1.0 m SLR 1.0 m
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
211
of certain structures such as breakwaters that become more easily overtopped by waves.
Such structures would have to be raised to maintain their effectiveness.
Sorenson et al. (2001) presented an overview of coastal engineering methods to control
shore erosion and inundation due to sea level rise. A number of methods can be employed
to control the effects of coastal erosion and shoreline inundation. These methods can be
classified as hard, soft and miscellaneous. Hard structures include; offshore breakwaters,
perched beach, groins, revetments, dikes, floodwalls, seawalls, bulkheads and dams. Soft
structures include; artificial beach nourishment, dune building and marsh building.
Miscellaneous responses include; elevation of structure and strengthening of the existing
structure. These control methods should consider economical, political and environmental
impacts on shore areas.
7.3.2.2 Control of saltwater intrusion
Over-pumping causes fall in groundwater levels below the mean sea levels in some areas.
Also, sea level rise accelerates the intrusion of saline water. Several control methods can
be used to prevent or retard saltwater intrusion into coastal aquifers such as; reduction of
abstraction from the aquifers, moving wells further inland, use of subsurface barriers to
prevent the inflow of seawater into the aquifers, increase the recharge to the aquifers either
by surface spread or injection wells to increase the volume of fresh water that retard the
intrusion of saltwater, abstraction of saline water to reduce the volume of saltwater or use a
combination of these methods. The selection of the appropriate method to control saltwater
intrusion should consider technical, economical and environmental aspects and sustainable
development of water resources in coastal areas (Abd-Elhamid and Javadi, 2010). Different
methods of controlling saltwater intrusion including a new methodology ADR presented in
this research were discussed in details in chapter 2.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
212
7.4 Simulation of saltwater intrusion considering sea level rise
The majority of the models developed to simulate and control saltwater intrusion do not
consider the effects of climate change and sea level rise in the simulation process. Only,
few researchers have considered the effects of climate change and sea level rise on
saltwater intrusion. EL Raey (1999) carried out an assessment of vulnerability and expected
socioeconomic losses in the Nile Delta coasts due to the impact of sea level rise of 50 cm
by 2100, in particular in Alexandria, Port Said, Egypt. Sherif and Singh (1999) investigated
the effects of likely climate change on seawater intrusion in the Nile Delta aquifer, Egypt,
and Madras aquifer, India. The study found that seawater intrusion is vulnerable to climate
change and sea level rise. Canning (2001) presented the certainties and uncertainties of
climate variability for Washington’s marine water, and suggested some areas of
management concern and research need. He has also presented a review to the historical sea
level rise due to climate change in Puget Sound. Oude-Essink and Schaars (2002)
developed a model to simulate groundwater flow, heat and salinity distribution, and
seepage and salt load flux to the surface water system. The model was used to assess the
effects of future development such as climate change, sea level rise, land subsidence as well
as human activities on qualitative and quantitative aspects of the groundwater system.
Tiruneh and Motz (2004) investigated the effect of sea level rise on the freshwater-
saltwater interface, using SEWAT. They coupled the impact of pumping with the rise in sea
levels and recharge based on selected climate change scenarios.
Also some models were developed to study the effects of sea level variations due to tides
on saltwater intrusion. Chen and Hsu (2004) developed a two-dimensional time-
independent finite difference model to simulate the tidal effects on the intrusion of seawater
in confined and unconfined aquifers. Mulrennan and Woodroffe (1998) examined the
pattern and process of saltwater intrusion into the coastal plains of the lower Mary River,
northern Territory, Australia, due to the effect of tides. Lazar et al. (2004) measured and
characterized the fresh water-saline water interface fluctuations by sea tides, by
measurements of electrical conductivity taken across the interface.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
213
Sonnenborg et al. (2010) presented an integrated and distributed hydrological model for a
coastal catchment on the island of Zealand, Denmark. They applied different scenarios for
different mean sea level rise in the range from 0.5-1.0 m. The rising seawater level resulted
in inflow of saltwater to the lake especially during late summer and autumn where the lake
water level is relatively low and storm events results in high sea levels. The study showed
that adaptation measures are needed already at a sea level rise of approximately 0.5 m if not
saltwater inflow should destroy the lake as a fresh water resource. Payne (2010) developed
a three dimensional finite element model for variable-density flow and solute-transport to
simulate different rates of sea-level rise and variation in onshore freshwater recharge in the
Hilton Head Island, South Carolina, USA. The results for this case showed that the greatest
effect on the existing saltwater plume occurred from reducing recharge and suggested
recharge may be a more important consideration in saltwater intrusion management than
estimated rates of sea level rise.
7.5 Application of SUFT to investigate the effect of sea level rise
In this work, the effect of sea level rise on saltwater intrusion is simulated using a density-
dependant finite element model. The development, validation and application of the
transient variable-density finite element model for the simulation of fluid flow and solute
transport in saturated/unsaturated soils was presented in chapters 3, 4 and 5. This part
presents the prediction of the intrusion of saline water in coastal aquifers due to likely
climate change and sea level rise. For the purpose of numerical modelling, a future rate of
sea level rise from 2000 to 2100 is considered based of the average measured rate of rise (1
cm/year) which represents a moderate expectation (IPCC, 2001). To investigate the effect
of sea level rise on saltwater intrusion the developed model is applied to a hypothetical case
study (Henry’s problem) and two real world cases studies in Biscayne aquifer, Florida,
USA and Gaza aquifer, Palestine.
7.5.1 Case (1): A hypothetical cases study
This case presents a hypothetical confined coastal aquifer which is developed based on the
well-known Henry’s problem (Henry, 1964). The aquifer is homogeneous and an isotropic.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
214
The aquifer is subjected to seawater intrusion along the sea face boundary and a uniformly
distributed lateral flow along the inland face. The geometry and initial and boundary
conditions and simulation parameters were presented in chapter 5. The dimensions of the
aquifer are 200 m (length), 100 m (depth). The domain is discretized using two dimensional
grid ( X= Y= 10m). The total number of elements is 200 elements and the total number of
nodes is 661 nodes. The flow boundary conditions include impermeable boundaries along
the top and the bottom of the domain. Hydrostatic pressure is assumed along the vertical
boundaries of the sea side and land side. At the inland side the concentration is zero
(freshwater condition), while at the coastal side the relative concentration of seawater is
imposed for a height 80 m from the aquifer bottom. Initial conditions are zero heads and
zero concentrations throughout the domain. The results of simulation using the above
mentioned conditions using the SUFT code, without consideration of sea level rise (SLR)
are shown Figure 7.5 which shows 0.5 isochlor in the aquifer.
Figure 7.5 0.5 isochlor for the hypothetical case study (no SLR)
The change in seawater levels are incorporated into the simulation model by specifying an
annual increase of 1 cm/year in the mean sea level. This rate implies a total increase of 100
cm by 2100. These simulations were carried out to assess the effects of the changing
saltwater boundary condition on water flow and the position of the fresh water/saltwater
interface in response to sea level rise. The saltwater intrusion in the aquifer is simulated
under different seawater levels; no sea level rise (static condition) and sea level rise after
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
215
25, 50, 75 and 100 years. The application of the simulation model to the current case has
been done on two steps. In the first step, the simulation model is applied with the initial and
boundary conditions to compute heads and salt concentrations at every node in the interred
domain with no change in seawater level. The distribution of salt concentrations without
changes in sea level is shown in Figure 7.5 which shows the 0.5 isochlor contour before
changing the water level on the sea side. In the second step, the simulation model is applied
to simulate the effect of sea level rise on water levels and the position and movement of the
fresh water/saltwater interface. The new position of fresh water/saltwater interface is
determined by a change in sea level by 1 cm/year from 2000 to 2100.
The results obtained from the simulation for the two steps are presented in Figures 7.5 and
7.6. In case 1 (no SLR), the 0.5 isochlor contour moved inland by 90 m measured at the
bottom boundary from the sea side as shown in Figure 7.5. In case 2 (SLR changing by 1
cm/year), the change in sea level affected the position of fresh water/saltwater interface and
moved the transition zone more inland as shown in Figure 7.6. The 0.5 isochlor contour
moved inland by 92 m after 25 years, 94 m after 50 year, 96 m after 75 year and 98 m after
100 year. The results indicate that the change of sea level has a significant effect on the
position of fresh water/saline water interface as shown in Figure 7.6.
Figure 7.6 0.5 isochlor for the hypothetical case study (with SLR)
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
216
7.5.2 Case (2): Biscayne aquifer, Florida, USA
In this case the developed simulation model is applied to a real case study to simulate
saltwater intrusion in a coastal aquifer considering sea level rise. Also, the effect of
lowering water table due to over-pumping is considered. The real world problem is selected
from the coastal area of Biscayne aquifer at Coulter area, Florida, USA. The aquifer has an
average depth of 30.0 m below mean sea level and width of 300.0 m. The domain is
discretized using two dimensional grid ( X= Y=6 m). The mesh consists of 250 elements
and 861 nodes. The schematic sketch of the Biscayne aquifer at Coulter area and the
corresponding boundary conditions were presented in chapter 5. The aquifer is simulated
using the developed finite element model and the results of simulation, without changes in
sea level are shown in Figure 7.7 in the form of 0.25, 0.5, 0.75 and 1.0 isochlor contours.
Figure 7.7 Isochlor contours in Biscayne aquifer, Florida (no SLR)
The change in seawater levels are incorporated into the simulation model by specifying an
annual increase of 1 cm/year in the mean sea level as a model boundary condition to assess
the effects of SLR on water flow and the position of the fresh water/saltwater interface. The
saltwater intrusion in the aquifer is simulated under different seawater levels; no sea level
rise (static condition) and sea level rise after 50 and 100 year. The distribution of salt
concentrations without changes in sea level is shown in Figure 7.7.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
217
The model is then applied to simulate the effect of sea level rise on the position and
movement of the fresh water/saltwater interface. The new position of fresh water/saltwater
interface is determined by changes in sea level by 50 and 100 cm after 50 and 100 years
respectively. The results of the new positions of the contours after 50 and 100 years are
shown in Figure 7.8.
Figure 7.8 Isochlor contours in Biscayne aquifer, Florida (with SLR, 50, 100 year)
To investigate the combined effect of sea level rise and lowering of water table due to over-
pumping on the position of fresh water/saltwater interface, three scenarios are considered.
In these scenarios, the shore line is maintained at its current location and the effect of
submergence of low lands by seawater is not considered.
Scenario 1; The water level in the sea is raised by 100 cm, the groundwater table is kept
unchanged and all the other parameters are kept to their original values.
Scenario 2; The piezometric head on the land side is lowered by 30 cm to investigate the
effect of additional pumping from the aquifer, the water level in the sea is kept
unchanged and all the other parameters are kept to their original values.
Scenario 3; The water level in the sea is raised by 100 cm, the piezometric head on the land
side is lowered by 30 cm and all the other parameters are kept to their original
values.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
218
The results of the three scenarios are shown in Figure 7.9 in the form of 0.5 isochlores for
the different scenarios after 100 years.
Figure 7.9 0.5 Isochlor contours in Biscayne aquifer, Florida using three scenarios
In the case where no sea level rise was considered, the 0.5 isochlor contour moved inland
by 135 m measured at the bottom boundary from the sea side as shown in Figure 7.7. In
scenario 1 (+100 cm SLR), the 0.5 isochlor contour moved inland by 142 m measured at
the bottom boundary from the sea side as shown in Figure 7.9. In this case the sea level rise
affected the position of fresh water/saltwater interface. In scenario 2 (-30 cm in water
table), the 0.5 isochlor contour moved inland by 139 m. it is shown that lowering the water
table or piezometric head on the land side due to additional pumping from the aquifer also
affected the position of fresh water/saltwater interface. However, the 0.5 isochlor contour in
scenario 3 moved inland by 145 m due to the combination of both sea level rise and
lowering water table.
Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
219
7.5.3 Case (3): Gaza Aquifer, Palestine
Another real world problem is addressed in this case, the coastal aquifer of Gaza Strip. The
developed model is applied to simulate saltwater intrusion in Gaza aquifer considering the
effect of sea level rise and lowering water table due to over-pumping. Three cross sections
at Jabalya, Wadi Gaza and Khan Yunis are simulated (Chapter 5) to assess the current
situation of seawater intrusion in the aquifer and the results are compared with the results
obtained by Qahman and Larabi (2006). The cross section at Jabalya is located in the north
of Gaza Strip with a length of 5000 m and average depth of 100 m. The domain is
discretized using two dimensional grid ( X=50 and Y=20 m). The mesh consists of 500
elements and 606 nodes. The boundary conditions applied in this case and model
parameters were presented in chapter 5. Gaza aquifer is simulated using the developed
finite element model and the results of simulation, without changes in sea level are shown
in Figure 7.10. This case was also simulated by Qahman and Larabi (2006) who used the
SEWAT code to simulate saltwater intrusion. The extent of seawater wedge along Jabalia is
about 3 km presented by isoline 2 (2000 ppm) which intruded inland to a distance of 3.0
km measured along the bottom boundary (see Figure 7.10).
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Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
220
In this case the effects of sea level rise and lowering water table due to over-pumping on
the position of fresh water/saltwater interface are investigated considering three scenarios.
In these scenarios, the shore line is maintained at its current location and the effect of
submergence of low lands by seawater is not considered. In the first scenario, the water
level in the sea is raised by 100 cm, the free water table is kept unchanged and all other
parameters are kept to their original values. In the second scenario, the piezometric head on
the land side is lowered by 0.5 m to investigate the effect of additional pumping from the
aquifer, the water level in the sea is kept unchanged and all other parameters are kept to
their basic values. In the third scenario, the water level in the sea is raised by 100 cm, the
piezometric head on the land side is lowered by 0.5 m and all other parameters are kept to
their basic values. The results of the three scenarios are shown in Figures 7.11, 7.12 and
7.13.
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Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
221
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Figure 7.12 Isolines of TDS concentration (kg/m3) along Jabalya (Scenario 2)
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Chapter (7) Effect of climate change and sea level rise on saltwater intrusion
222
In the first scenario (+1 m SLR), isoline 2 intruded inland to a distance of about 3.5 km
measured at the bottom boundary from the sea side. In the second scenario (-0.5 m lower in
water table), isoline 2 intruded inland to a distance of about 3.25 km. However, isoline 2 in
the third scenario intruded inland to a distance of about 4.0 km, due to the combination of
both sea level rise and lowering water table. The Gaza aquifer is nearly contaminated by
saline water and urgent actions are needed to prevent such intrusion because groundwater is
the only source of water for about 1.5 million Palestinian living in Gaza strip.
7.6 Summary
The rate of climate change is increasing rapidly due to human activities. Human societies
and natural ecosystems may not be able to adapt to rapid climate change. The increase in
the global temperature will warm the land surface, oceans and seas and melt the ice caps,
which will in turn lead to the increase in water levels in the oceans and seas. The rise in
water levels could have damaging effects on residents of coastal areas because of higher
rates of erosion, inundation and saltwater intrusion. The rise in water levels may lead to
partial or full submergence of some coastal cities which lie in low lands. Sea level rise also
threatens groundwater storage in coastal aquifers as it imposes an additional pressure head
at the seaside which accelerates saltwater intrusion. The study of the effects of climate
change and sea level rise on coastal areas has become an important subject. In this study the
effect of climate change and sea level rise on coastal aquifers is studied using the developed
numerical model. The results of the model indicate that the sea level rise has a significant
effect on the position of fresh water/saltwater interface. Furthermore, lowering of water
table or piezometric head due to additional pumping from the aquifer also affects the
position of fresh water/saltwater interface. The combination of both sea level rise and
lowering water table has a greater effect on the position on fresh water/saltwater interface.
Chapter (8) Conclusions and recommendations
223
Conclusions and recommendations
8.1 Conclusions
Seawater intrusion represents a great risk to groundwater resources in coastal areas as it
raises the salinity to levels exceeding acceptable drinking water standards. Salinization of
groundwater is considered one of the main pollutants in coastal aquifers. Moreover,
climate-change predictions suggest that as temperatures rise, the droughts would
exacerbate, and as the sea level rises, saltwater intrusion would speed up. Rising sea level
poses another challenge to the coastal areas as it could lead to the submergence of much of
coastal cities in different parts of the world. Although some research has been done to
investigate saltwater intrusion however, only a limited amount of work has concentrated on
the control of saltwater intrusion to protect groundwater resources in coastal areas which
represent the most densely populated areas in the world, where 70% of the world’s
population live.
This work presented a new methodology (ADR) to control saltwater intrusion into coastal
aquifers. Two main approaches were used in this study; the first approach was the
development of a coupled transient finite element model of fluid flow and solute transport
in porous media to simulate saltwater intrusion and the second was the development of a
simulation-optimization model to control saltwater intrusion in coastal aquifers using
different management scenarios. The development of the numerical model for predicting
solute transport through saturated/unsaturated soils is a major theme considered in this
work. The presented simulation-optimization model to control saltwater intrusion is another
main theme considered in this work. The application of the proposed methodology ADR to
control saltwater intrusion is a major achievement of this work. Also the application of the
developed model to simulate the effect of sea level rise due to climate change is another
achievement.
Chapter (8) Conclusions and recommendations
224
The achieved objectives in this work include:
A coupled fluid flow and solute transport model was developed to solve the
governing equation of solute transport together with three balance equations for
water and air flow with heat transfer which provide values of solute concentration,
pore water pressure, pore air pressure and temperature at different points within the
flow region at different times.
The numerical model was validated by application to several standard examples
from literature followed by application to a number of case studies involving
seawater intrusion in coastal aquifers.
The numerical results illustrated the performance of the presented model to handle a
wide range of real problems including the simulation of fluid flow alone, solute
transport alone and coupled fluid flow and solute transport.
The results showed that the developed numerical model is capable of predicting,
with a good accuracy, the movement and extension of saltwater in coastal aquifers.
The developed model for simulating saltwater intrusion considers transient, density-
dependent flow in saturated/unsaturated porous media. Dispersion is considered to
be velocity-dependant. Also, different mechanisms which affect solute transport in
porous media including, advection, diffusion, dispersion, adsorption, chemical
reactions and biological degradation are considered.
A simulation-optimization model was developed for the control of saltwater
intrusion in coastal aquifers using three management scenarios including;
abstraction of brackish water, recharge of fresh water and combination of
abstraction and recharge. The objectives of these management scenarios include
minimizing the total costs for construction and operation, minimising salt
concentrations in the aquifer and determining the optimal depths, locations and
abstraction/recharge rates for the wells.
A new methodology (ADR) to control seawater intrusion in coastal aquifers by
abstracting brackish water, desalinating it using small scale RO and recharging to
the aquifer was presented. The results showed that this methodology is an efficient
and cost-effective method to control seawater intrusion in coastal aquifers.
The developed model was applied to study the effects of the likely climate change
and sea level rise and/or over-pumping on saltwater intrusion in coastal aquifers.
Chapter (8) Conclusions and recommendations
225
8.2 Recommendations for future work
This study has touched a serious real world problem which has become of considerable
concern in many countries with coastal areas. It presented a first step in developing a
simulation-optimization model to control saltwater intrusion in coastal aquifers. This work
presents a new methodology ADR to control saltwater intrusion in coastal aquifers through
the application of a simulation-optimization model. As has been discussed there are several
benefits for using ADR methodology to control saltwater intrusion however this work is
regarded as a first step in developing simulation-optimization models and more work can
be done in the future. A list of possible topics for future work is as follows:
Extend the current model to combine horizontal simulation with vertical simulation
to obtain a 3D picture of seawater intrusion and determine the distance between
wells and the number of wells required to control SWI using ADR.
Extend the current model from 2D to 3D, which would give more realistic
simulation for practical applications of most of saltwater intrusion problems.
More work needs to be carried out on the experimental study of saltwater intrusion
in coastal aquifers and comparison with numerical results.
The effect of temperature on the intrusion of saltwater and the flow in vadoze zone
also needs further investigation.
Integrating the developed model with a decision support system for sustainable
development of water resources management in coastal areas is another possibility
for future work.
References
226
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